The Pyro is a class in TF2 frequently called "OP". However, the Pyro has many counters and weaknesses.
The Pyro has 175 health (overheals to 260) - more health than a sniper, spy, engineer, and medic, and equal health to a demoman, but less health than a heavy and soldier. He travels at 100% speed - faster than a heavy and soldier, slightly faster than a demoman, equally fast to a sniper, spy, and engineer and slightly slower than a medic and slower than a scout.
First of all: A small Introduction to the pyro's playing style.
A Pyro has 3 weapons - A Flamethrower (Primary), a flaregun/shotgun (secondary), and a melee.
The flamethrower has a very limited range and does massive damage at close range, forcing the Pyro to mainly rely on ambushing and suprise attacks. The fire also goes through multiple targets, allowing the Pyro to hit multiple people easily. The flamthrower also has a unique secondary attack - the compression blast. The compression blast, or airblast, launches enemies in the air, and reflects projectiles such as grenades, stickies and rockets. A skilled Pyro can easily clear a stickied area, or airblast grenades and rockets back. When a Pyro reflects a projectile, they deal mini-crits (35% extra damage). Beware when shooting slow projectiles at a Pyro. The flare gun is a ranged weapon for the pyro. It shoots a projectile that is slow-ish and arcs, making it hard for the Pyro to actually hit anyone at long range that is moving. It is used by the Pyro to harass or finish off weakened targets. The shotgun is a medium ranged hitscan weapon, firing a burst of bullets. Similarily, it is used to harass or finish off weakened targets. Some Pyros use the shotguns and flare guns offensively, after airblasting an enemy in the air. Melee for the pyro is different for each melee. Each Pyro melee is unique and has different properties.
When fighting a Pyro, always have a backup plan. If if your backup plan fails, always have a plan C. If you know you are outskilled or simply cannot defeat the Pyro, find a way to distract the Pyro long enough for you to get help from a teammate or get away.

The Pyro has many unique weapons at its/her/his disposal.

PRIMARY WEAPONS
The Stock Flamethrower
The stock flamethrower is the regular flamethrower. It has a short range, but deals large amounts of damage at close range. It sets enemies (except other Pyros) on fire, causing afterburn, which does damage over time. The Flamethrower can also airblast, pushing back enemies and reflecting projectiles.
Countering it: Try to stay out of the reach of a flamethrower. It has a short range, so if a Pyro is charging at you backpedal and shoot constantly at the Pyro. You can also try and plain run away, but watch out for Pyros who cut corners and use their secondary to take you down. You can also rocket or sticky jump away if needed.Do not run towards the Pyro, unless you a. are suicidal, b. have balls of steel. If a Pyro ambushes you, try to run away, but do not turn your back. Don't panic, and shoot at the Pyro. Try to take him down or weaken him so your team mates will have an easy time with it/her/him. Note where med packs, water, etc. are so you can extinguish yourself as fast as possible. Use hitscan weapons to rip the Pyro to shreds, and be careful when using projectiles. If the Pyro is already in close range, and you know the Pyro has enough health to survive a shot from your primary, you could go in for a melee. Melee has a higher chance of crit-ting (15% - 65% chance of crit-ting, depending on the damage you've done in the last 20 seconds), and a melee crit will usually take a Pyro down (most melee crits are 195 damage).These rules also apply to the other flamethrowers unless otherwise stated.

The Backburner
The Backburner is a flamethrower designed for ambush. It costs 50 ammo for the Backburner to airblast (compared to the normal cost of 20 ammo), but deals crits(3 times normal damage) from behind.
Countering it: Watch your surroundings carefully, especially if you are on a very roundabout map. The Backburner crit radius isn't very large, so if you turn to your sides, it won't crit you. DO NOT run from one, unless you are a scout or you want to become barbequed. Because of the Backburner's extra airblast ammo usage, you can more safely use projectiles.

Phlogistinator
The Phlogistinator is a flamethrower designed for agressive W+M1 longetivity and mass murder. It has no airblast, but once the Pyro deals a certain amount of damage (225 points of damage), the Pyro can taunt (while taunting, the Pyro has a 75% damage resistance) and fully heal it/her/himself, then recieve crits for 10 seconds.
Countering it: Blast the Pyro to pieces with projectiles and whatever else you want. The Phlogistinator cannot reflect projectiles, and projectiles are medium to long range, so the Pyro can't do much about them. When it/he/she is taunting, attack at a distance (the damage resistance does not apply to backstabs, you can stab a Phlogistinator pyro), or just set up a kill right after the Pyro finishes taunting (stickies). When he/she/it recieves crits, stay at a distance and attack.

Degreaser
The Degreaser is a flamethrower designed for quick combos, and is very popular among advanced Pyros.. It allows a Pyro to switch his/her/it's weapons 65% faster, at the cost of doing 10% less damage and 25% less afterburn damage. Because the Degreaser does slightly less damage but has a faster weapon switch, most Pyros using the Degreaser use comboes to kill people. A common combo, the "Puff-and-Sting" Pyro, is a Pyro using the Axtinguisher to crit people on fire. The Puff-and-Sting Pyro will shoot a puff of fire, and then either airblast or directly crit (sting) the burning enemy. Many Degreaser Pyros will also use a Reserve Shooter, Flare Gun, or Powerjack.
Countering it: This style of Pyro is very close range, and relies on the airblast. Take the pyro down before it can hit you. Avoid cramped areas to reduce the effectiveness of the combo.

SECONDARY WEAPONS

Shotgun
The Shotgun is a familiar secondary weapon. It is hitscan and shoots a net of bullets, allowing the Pyro to easily finish an enemy off at medium range.
Countering it: Although a shotgun is medium range, many other classes are more effective at medium range than a Pyro with a shotgun is. A shotgun also cannot airblast, so if a Pyro pulls out a shotgun, it is safe to use rockets or grenades.

Reserve Shooter
The Reserve Shooter is a shotgun that helps a Pyro is combos. It boosts the Pyro's weapons switch 15%, has 1/2 the clip a shotgun does, and mini-crits an airborne enemy 3 seconds after switching to it. It is used in conjunction with the Degreaser, so the Pyro can manually put an enemy in the air for mini-crits.
Countering it: Don't jump around the Pyro. Stay on the ground. Stay a distance away, and remember: it has a smaller clip. If the Pyro has had it out for a few seconds, you can jump and leave the Pyro wondering what's wrong with his gun.

Flare Gun
The Flare Gun is a completely different class from the shotgun. It shoots a mildly fast (only faster than arrows (most of the time), cleavers, bat launched projectiles, rescue ranger shots, and fully charged sticky shots), arcing projectile that crits burning people. It only does 30 damage, but when it crits, it deals 90 damage. It also sets people on fire. A Pyro uses this for harassment or finishing people off.
Countering it: Move around. It is quite fast, but the projectile is small, so it isn't too hard to dodge it. Don't move in a straight line, and remember that it arcs. Don't go out in the open if you are on fire.

Detonator
The Detonator is a flare gun that detonates (explodes) when the Pyro uses it's/her/his secondary fire. It can hit multiple people, and teh detonation can allow a Pyro to soar through the air a little (not nearly as high as a rocket jump) if shot at the Pyro's feet.. It also only deals mini crits (41 damage)on burning enemies.
Countering it: Don't stay in a bunch. Watch out, as a Pyro has extra mobility. Be extra vigilant, as the Pyro can reach places it/he/she can't normally reach. Dodging is harder.

Scorch Shot
The Scorch Shot is a flare gun the pushes enemies back and does 50% damage. The push-back grows as the Pyro is farther and farther, and can push people very far. The flare has a unique property of dropping to the ground after hitting someone, and anyone else who gets hit be the dropping/dropped flare will also ignite. It does a feeble 15 damage, however. The Scorch Shot is a super harrass-er. Can also jump, but not as high as a detonator jump.
Countering it: Don't stay close to environmental hazards, like falls and pits and sawblades. A Pyro can repetitively push you back, so try and dodge the flare and circle around the Pyro. Don't stay in bunches.

Manmelter
The Manmelter is a flare gun that can extinguish burning teammates (each teammate extinguished gain a crit on the Manmelter), does not need ammo, does not have random crits, and the projectile flies 50% faster. Beware if your Pyros are leaving a bunch of burning enemies alive.
Countering it: Make sure your Pyros don't just set people on fire and leave. It is a bit harder to dodge, but less useful for combos. Make sure the Pyro doesn't have many crits. Harass the Pyro to shoot a flare at you and dodge it or stay near a med-pack, forcing them to waste their crits.

I think I maxed out on characters in the last section, so Pyro melees will be described in a seperate section.

MELEE

Fire Axe
You will almost never see a Pyro use this. When you do, the Pyro is probably desperate and out of ammo.
Countering it: Just kill the Pyro out of the reach of it's/his/her melee range.

Axtinguisher
A very dangerous axe that deals crits on burning targets, but deals 50% less damage on normal targets. Used almost all the time with the Degreaser, to Puff-and-Sting.
Countering it: Keep the Pyro far away, and keep yourself not burn-y. Try and use knockback to keep the Pyro away, because an Axtinguisher crit + a second of afterburn is enough to finish off everything except a heavy (not including overheal).

Homewrecker
A hammer that is a support melee, and rarely used in direct combat. It does 25% less damage on players, but it remove sappers and does double damage on enemy buildings.
Countering it: The only classes that should worry about this are the spy and the combat engineer, to an extent. As a spy, you can't do much to defeat a stubborn pybro. Try to harass (especially if you have an Ambassador) the Pyro and get it to chase you, and then set up a trick stab or gun him down. Mini sentries can usually deal a large amount of damage before the Pyro can destroy it. One or two shotgun blasts should finish the Pyro.

Backscratcher
A rake that the Pyro uses to scratch his back. Making the Pyro's back itchy is ineffective, so do not do it anymore.
Okay, for real: The Backscratcher gives the Pyro a 50% boost of health from med-packs, and deals 25% extra damage, but reduces healing from medics and dispensers by 75%. Roaming Pyros use this melee.
Countering it: Don't let him get to close, and close off access to med-packs.

Sharpened Volcano Fragment
A vastly unpopular axe that looks cool - but is rarely used. It does 20% less damage, but sets enemies on fire.
Countering it: Don't let the Pyro get close, blah blah blah, you've heard it before.

Third Degree
A pretty axe that hits everyone connected be heal beams. So if you have a medic train, and a Pyro hits the guy in front, everyone gets hit. Pyros usually hit a heavy and try to kill the medic.
Countering it: Cut the beam off and try to take down the Pyro with your buddy. The only class this is used against is the medic.

Neon Annihilator
A neon sign that does 20% damage less, but crits wet enemies and has no random crits. Also removes sappers. Pyroshark will get you.
Countering it: Try to stay out of water. If the map has rain, ouch. Don't let the Pyro get too close

Powerjack
A hammer that gives the Pyro the ability to heal 75 health with he/she/it gets a kill, but takes 20% more damage while it is out. The Pyro can also run 15% faster with it out.
Countering it: Normally, the Pyro will use the Powerjack to get a quick kill, or to roam around. Watch out, as the Pyros may give chase unexpectedly, or be able to get to ambush spots more quickly. They'll be able to run pretty quickly, so try to remember that. If they attempt to puff n' sting you, try to get some hits on them, as it probably won't be an immediate kill unless your health is already down.

One of the Pyro's unique abilities is to cause afterburn. Afterburn is inflicted when the Pyro hits an enemy with a flamethrower, flare gun, or the Sharpened Volcano Fragment. The Pyro can use afterburn to harass enemies and force them to run away to extinguish themselves. WHen you are set on fire your class will automatically scream that they are on fire, and if youc all for medic, the medic bubble will be surrounded in flames. Flames also appear on the side of your scream, and the screen flinches a bit each time, making it useful to disrupt precise aiming, like with a sniper. Fire lasts for 10 seconds, dealing 60 damage in that amount of time. Degreaser afterburn only does 40 damage in 10 seconds. Afterburn doesn't crit, but it can mini-crit if you have the GRU out, or you are soaked in Jarate, etc. Pyros cannot be set on fire, although for a brief time, they count as on fire.
Countering it: The most common methods of extinguishing yourself would be to find a med-pack, immerse yourself in a body of water (it can't be a puddle), having a friendly Pyro airblast you, finding a medic, or a dispenser. Snipers and scouts can also toss their Jarate and Mad Milk to extinguish a burning player, and being Ubercharged witht the stock medi-gun will remove the fire. A Manmelter can also extinguish teammates.
Spies beware of afterburn: a Pyro and his/her/it's teammates can easily track you down if you are on fire. A Dead Ringer can extinguish the fire, but a Pyro can still easily hit you afterwards when you are cloaked and airblast you into a corner or follow you. Switching to the Spy-Cicle can also extinguish a burning spy, and backstabbing someone with the Conniver's Kunai will. If oyu do burst into flames, try and shoot the Pyro with your revolver - at close range, it should take three to four shots, and if you use an Ambassador, you can headshot them for more damage.

The compression blast, also known as the airblast, is another unique skill the Pyro has. The airblast shoots out air, costing the 20 ammo (or 50 ammo with the Backburner). The airblast pushes enemies back, extinguishes teammates, and reflects certain projectiles. Rockets, grenades, Stickybombs, the two bat-flung projectiles the scout uses, thrown liquids, flares, arrows, cleavers, and the Rescue Ranger claw can all be reflected. When these are reflected, they become mini-crits. The only reflects that a Pyro can pull off consistently are the sodlier's rockets and the demoman's grenades and stickies. A Pyro can also airblast a burning enemy inot a corner to trap and disorient them.
Countering it: To counter the airblast, don't let the Pyro get too close. You are still alive while in the air (airblasts do no damage), so keep shooting the Pyro.
Soldiers can "weapons heckle" the Pyro - use the shotgun at medium range so the Pyro will take out his/her/it's shotgun or flare gun, then shoot a rocket before the Pyro can react. The Pyro will then switch back to his/her/it's flamethrower, and then you can shoot it/him/her with the shotguna nd repeat.
Demomen have a harder time, since he has no hitscan weapons. A Loch-n-Load is pretty good for hitting a Pyro directly. Try to put stickies around a corner or above an entrance and lure the Pyro in and then detonate the stickies, or sticky jump away if you can't. Try to peg the Pyro with a direct hit from your grenades.
When the Pyro airblasts one of your projectiles, remember it is a mini-crit, and deals more damage.

The Pyro is a very short ranged hit-and-run ambush class. Be aware of your surroundings, and know the areas of med-packs, dispensers, medics, etc. He can also reflect projectiles, making him/her/it a dangerous adversary when you are a soldier or demoman. Fire can pass through multiple targets, so do not stay in bucnhes.

Average Flamethrower Pyro
A Pyro that relies mostly on their flamethrower. Try not to bunch up too much, to lessen the effectiveness of the flamethrower, and be aware for ambushes.

Puff-and-Sting
The Puff and Sting Pyro is a very dangerous Pyro. Usually, a puff-and-sting Pyro uses the Degreaser, Flare Gun/Reserve Shooter, and the Axtinguisher. A Puff-and-Sting Pyro usually puffs a bit of fire at you, airblasts you, then Flare Guns/Reserve Shooter/Axtinguishes you.
Countering it: If you know the p-and-s Pyro is using a Reserve Shooter, don't jump while fighting the Pyro. Keep out his/her/it's range, and keep on attacking the Pyro when he/her/it airblssts you.
A competitive puff-and-sting Pyro will rely on their secondary for a ranged stinged rather than a close up melee sting most of the time, unless they are ambushing you. Their strategies will differ depending on their secondary. A puff-and-sting shotgun Pyro will be able to stay in combat a long time, due to the shotgun having a large clip. A Reserve Shooter Pyro will probably try to kill you quickly, but won't have as much staying power due to a smaller clip, and the Reserve Shooter is more reliant on the airblast than a shotgun. A Flare Gun Pyro will try to get a crit flare, but if that misses, they'll switch to their flamethrower for a bit to take advantage of the Flare Gun's passive reload.

Pybro
A Pybro is a Pyro that guards an Engineer's sentry. A Pybro usually uses the Degreaser or Flamethrower, and the Homewrecker. A sentry is weak to spies and explosive projectiles, all of which the Pyro and effective against. A Pybro can protect a sentry for quite a while.
As a soldier or demoman, try to use the loch-n-load and direct hit to take out the Pyro first, because it is harder for the Pyro to airblast these. As a spy, try to lure the Pyro out, or harass him/her/it with a revolver, preferably an Ambassador.

Pyroshark
A Pyroshark is a rare type of Pyro that uses the Neon Annihilator. They lurk in the murky water, waiting for a quick dinner (kill) with the Neon Annihilator's easy crit in the water.
Countering it: Stay out of the water. If you are in the water, youarescrewedrun , try to not let the Pyro get to close. A Neon Annihilator crit does ~150 damage, so it won't kill a Pyro, demoman, soldier, or heavy immediately (not including overheal).

Pyromania
A Pyromania Pyro would be a Pyro with the set Rainblower, Scorch Shot, and Lollichop. The Rainblower and Lollichop are both reskins of the stock flamethrower and fire axe, respectively. This set means they are using Pyrovision - Pyrovision is where certain maps become very pink-ified, representing the magical world the Pyro believes in. A Pyromania Pyro will rely on the Rainblower to attack and ambush, and the Scorch Shot to harass. Be aware of your surroundings, and beware when attacking the Pyromania Pyro with projectiles.

How the Pyro scales up against the other classes.

Scout
A Pyro has a hard time dealing with scouts, unless the two are in a cramped area. Circle strafe and attack the Pyro from medium range. You can easily get away if you are losing; find a med-pack. You can also Bonk! and run away (don't let the Pyro airblast you) and use Mad Milk to extinguish yourself. The polycount set Milkman was (I think) specifically designed to destory Pyros. The Shortstop can blast the Pyro at medium range, the Mad Milk can extinguish the scout, the Holy Mackerel is a reskin of the stock bat, but you can humiliate the Pyro with it, and the extra 25 health helps increase the scout's survivability. Make use of your double jumping too. Use your pistol if you really have too. You can also stay at medium range and wait for teh Pyro to switch to their secondary, and come in quickly to blast them with your scattergun before they can switch back to the primary, and repeat.

Soldier
Try to stay up high, and make good use of rocket jumping. You can weapons heckle the Pyro - use your shotgun at medium range to harass the Pyro. The Pyro will likely switch to their secondary, and then you can hit the Pyro with a quick rocket. Switch back to the shotgun once they switch to their flamethrower, and repeat. Weapons heckling might not work as well with skilled Degreaser Pyros. You can also aim your rockets at places near the Pyro, so it/he/she can't quite reflect it well, but the splash damage will hurt the Pyro. Try and bomb the Pyro from above so its harder for the Pyro to reach you, and the Pyro cannot effectively airblast a rocket against you.The Direct Hit and Beggar's Bazooka can also be effective - the faster speed of the rocket and the burst of sudden rockets can be effective for countering a Pyro. Watch out for super-pyros that use your rocket to fly over to you. Be prepared for this, and remember that the Pyro takes damage from the jump. Try to get away or finish the Pyro off before he/she/it gets too close.

Pyro
Use your airblast and shotgun to full use. Shoot fire while backpedaling, as your flames with travel farther the the enemy Pyro advancing. Circle strafe while burning the Pyro. The Reserve Shooter is particularly effective against enemy Pyros. Try to make use of your Backburner or Phlogistinator crits if possible.

Demoman
Try to lure the into a sticky trap, or directly hit the Pyro with a grenade (Loch-n-Load for mazimum effectiveness). You can also detonate your stickies before a Pyro can get into airblasting range, and the demoman's grenade launcher shoots faster than a Pyro can airblast. You can charge your stickybomb launcher to shoot stickies faster than normal. If you are a demoknight, try and ambush the Pyro and take him/her/it down with a shield bash and crit swing. Take note that a Pyro can cancel your charge by airblasting, and the Axtinguisher is not affected by your fire resistances. A Pyro usually has the upper hand if you are a full demoknight. However, the Chargin' Targe can be quite effective against Pyros. It nullifies afterburn and had a 50% fire resistance, so the Pyro cannot puff n' sting you, and it'll take significantly longer for them to kill you. Try to use your Grenade Launcher as much as you can if a Pyro catches you as a Demoknight.

Heavy
Rip him/her/it to shreds with your minigun. If you are ambushed, you'll still stay alive for quite a while. Keep the Pyro at medium range. The Natascha is somewhat effective at slowing down the Pyro, making it harder for the Pyro to approach or circle strafe you.

Engineer
A sentry can also rip a Pyro to shreds, and you can't ambush a sentry. If your sentry is right around a corner, the Pyro can shoot the flames at an angle to hit the sentry while staying out of sight. If the Pyro can get close enough, he/she/it will attempt to circle strafe the sentry. Shoot the Pyro with a shotgun or pistol. Don't let the Pyro ambush you while you are hauling a building. Keep dispensers in easy-to-access places so your teammates can easily extinguish themselves.
If you are alone and without a sentry, use the Pistol and Shotgun or Frontier Justice crits to fend him/her/it off.

Medic
Be aware of your surroundings. If a Pyro ambushes or chases you, backpedal while shooting him/her/it with your syringe gun. This is most effective with the Overdose and Blutsauger. Be wary of a Pyro flare jumping to you.

Sniper
Snipe the Pyro at long range. If he/she/it gets up close, use the jarate-bushwacka combo to take him down with a swing, or use the SMG in close quarters. If you prioritize your survivability over your kill-ability, then you can switch out the bushwacka for a stock kukri (for normal damage) or tribalman's shiv (to cause bleed), because the bushwacka has a 20% fire vulnerability.You can also use Jarate to extinguish yourself. Move around and stay covered so the Pyro has a harder time hitting you with a flare. If the Pyro is using the normal Flare Gun, if you get hit by two flares in succession without extinguishing yourself, its lethal. If you are on fire, you will start to flinch, throwing off your aim. Watch out for Pyros flare jumping to you, especially if you are on a lowish platform.

Spy
Be wary of the Pyro. Perform a trickstab (a simple one is the aerial stab: Find a high platform, a drop down behind the Pyro when he walks along. People almost never look up.). If the Pyro is chasing you, shoot him or cloak and run away. Spy-cicle is more effective at combating Pyros; you can use the extinguishing time to shoot the Pyro or cloak and run away. The Enforcer and Ambassador can be useful for killing the Pyro in combat, and the L'Etranger can help you charge up your cloak (which helps with Dead Ringer, but when using DR, beware of a Pyro sweeping the area with fire once you're dead).

Now you know how to kill a Pyro. Congratulations. Feel free to tell me any feedback - negative or postitive.

Let me know if I have any typoes, and if you have more tips and tricks, tell me (you can add me if you like) and I'll put it in here with your name on it. If you need pictures of what the weapons look like, go here http://wiki.teamfortress.com/wiki/Pyro . Its the official TF2 wiki.

Credits~
Pyro overheal information corrected by Indivicivet // back to TF2!.
Using explosive jumps to get away inspired by ЩĄ | Jaden.
Using melee as a last resort suggested by Tolna.
Flare speed corrected by Indivicivet.
Mini crit info with the detonator made more accurate by Indivicivet.
Whole Powerjack section reminded to me by Shiew and The Fabulous Lactating Yeti.
Competetive puff-and-sting Pyro strategies taught by Mr. Chippy.
Pyromania Pyros suggested by DarkCrystalMethod
Using rocket splash damage against Pyros stated by ▫□☐GnomePuncher☐□▫
Grenade attack intervals credit to ROYAL_CHAIRFORCE).
Demoknight ineffectiveness debated by {ZBS} Kinglegos,.
Alternatives to the Bushwacka given by Kingsania.
Flare jumping and flinching noted by lucas_2400
Demoman stickies, soldier stuff, backup plans, and airblasting all thanks to Firebrand!
Combos typos ("comboes") corrected by Pet.
More Spy items added in thanks to Zombiebutcher.
Gas Jockey stuffs added in thanks to ShadowKnight - (No Sound)
Updated for the big update thanks to [EC] FredZed