In SHOGUN 2 there are four distinct unit tiers that almost all units can be put into. These are:

Ashigaru

Ashigaru are quickly trained and inexperienced commoners who were ordered off their land to fight for their clan. They have low morale, low armour, low attack and low defence. Because of this, they are generally useless and the only useful ashigaru unit is yari ashigaru.

Warrior Monks

Warrior Monks are religiously fueled ferocious warriors with high morale, attack and defence, with the trade off being their low armour. Warrior monks are more skilled than other units, often giving them field advantages such as longer firing ranges for bows and the borderline overpowered warcry ability.

Samurai

Samurai are rich noble men who believe their mission in life is to die. They have quite decent morale, attack and defence and have incredibally high armour. Samurai make up the majority of most armies. All cavalry are mounted by samurai.

Heroes/Heroines

Hero units are master soldiers trained in bushido with incredibally high attack, defence and nearly unbreakable morale, however have samurai level armour. This makes them nearly unbeatable in melee combat, but very vulnerable to missile attacks. Skill cannot compensate for armour.

Obviously soldiers have weapons to murder and disembowel their foes, but what weapons are there in SHOGUN 2?

Yari

The yari is a long shaft of wood, metal or bamboo with a narrow sharp piercing head on the end designed for lethal thrusting and piercing. It is incredibally effective against cavalry units, often dropping 50 without losing a single man in a unit. However, the yari is incredibally ineffective in infantry combat, usually just being effective tank units.

Katana

The katana is a razor sharp cutting edge sword requiring great skill and patience to make. Katanas were often considered family heirlooms, considering their great meaning and value. But what if it was used in a battle?
Katanas are the most powerful infantry vs infantry weapons in SHOGUN 2. They are clearly very effective at murdering other soldiers.

No-dachi

The no-dachi is a large, frontal heavy sword with an intimidating sized blade. No-dachi are incredibally effective at charging, often killing 50-80 men in a charge - DAYUM!

Naginata

The naginata was designed as a compromise between the yari and the katana. It is a long cutting edged blade mounted on a short pole, giving the wielder a distance advantage as well as a cutting and piercing weapon which could both take on cavalry and infantry. In-game it is effectivelly a weaker katana unit that can somewhat defend itself against cavalry. Naginata samurai have rediculously high armour, making them the best meatshields in the game and a good front line to tank both missile fire and cavalry banzai charges.

Yumi Bow

The yumi bow is a medium sized bow capable of firing large distances and used from horseback. The yumi bow is very effective at shredding low armoured units such as monks and ashigaru, quickly leaving a trail of bodies behind a charging unit. Any unit under fire from yumi bows will ungratefully recieve a morale hit.

Matchlock Rifle

Matchlock rifles are primative, slow firing short range rifles imported from the "east" that are just about unstoppable. Units being shot at with matchlock rifles suffer a massive morale shock.
They are deadly weapons in the hands of skilled soldiers, however are hindered by their long reload times, short range and requirement of a clear view of the targeted enemy unit, as unlike yumi bows they cannot be arced through the sky over terrain and other units.

Cavalry

Cavalry are agressive soldiers on fast moving horses. Cavalry has high charge bonuses and can easily kill katanas and missile units in melee combat. As with the other units, I will go into detail about them below, as cavalry is a very diverse group.

Your general is arguably the most fragile but important unit in your army. He gives a morale boost for local units and can provide powerful buffs such as inspire and rally.
Keeping your general alive is important, as once he dies your army will recieve a devistating morale shock and you may as well just surrender unless the opponent has also lost his general, as once in combat, your units will probably instantly run from the battlefield.

Yari ashigaru are ineffective infantry fighters, who however do a fantastic job against cavalry, are cheap, in large numbers and have a special ability called yari wall, which puts the unit into a tight wall and makes them deploy their spears forwards. This ability slightly raises the defence of the unit, and reflects the charge bonus of an enemy unit back at itself. So if a katana samurai unit were to charge into a yari wall, the katana unit would suffer the damage it would have given to the yari unit if it had not been in yari wall. This makes this ability invaluable against cavalry.

Yari samurai are a more effective, tankier version of yari ashigaru, minus their amazing yari wall ability which is replaced with the rapid advance ability.
In my opinion, yari samurai should NEVER be used. They are expensive trash.

Naginata samurai are tanky effective warriors with ludicrously high armour. They can take on cavalry some-what effectively and hide from arrows under their metal coats. They are good for army front lines, because as I said above they can defend against cavalry and also tank arrows.

Naginata warrior monks are deadly melee fighters with the borderline overpowered war cry ability which lowers local enemy morale. They hit like a truck, but take heavy missile responsible casualties due to their paper armour.

Lone-sword Ashigaru are discount katana samurai with low armour. They are very cost effective once upgraded in the veteran system, however before that point they are very ineffective, even being beaten by naginata units. If you have the time, patience and nerves to get these guys to high veteran levels, then you have yourself incredibally effective killing machines, however if that isn't you, then you may as well look past them.

Katana samurai are in my opinion the best melee fighters in SHOGUN 2. They will obliterate any other unit in melee nearly effortlessly, and their decent armour allows them to take a considerable amount of shots without significant casualties. Their only weakness like other sword units are cavalry. I highly recommend using katana samurai.

No-dachi samurai are semi-low armoured samurai units that while somewhat effective in infantry combat, truely shine in their charge bonus, which is the highest in the game, even higher than cavalry! To be honest, I rarely use no-dachi samurai however in the right build, they can be extremely effective. Also, no-dachi samurai are very effective against cavalry for some reason... If you charge with a no-dachi unit against a charging cavalry unit, chances are 30-40 of the cavalry will have died within two seconds.

Bow ashigaru are actually the most useless units in SHOGUN 2. I legitimately could complain about them for hours. They have a slow fire rate, low armour, low ammunition levels and often rout without losing a single man. They CANNOT obtain special arrows such as flaming arrows and whistling arrows, which are key tacticle abilities. Do yourself a favour and slap yourself on the wrist if you ever feel like using these "units".

Bow samurai are effective infantry shredding machines that can easily drop hundreds of low armoured units such as ashigaru and warrior monks. I simply love them. If engaged in melee combat, they are able to hold their ground for a while so you can reinforce the position.

Bow warrior monks are effectively low armour bow samurai, but with a slightly longer range (which can be seriously abused in castle seige mode) and a faster fire rate. These are good units, however they are very expensive and in my opinion it is better to instead get a bow general and/or the mounted bow hero unit, both of which outrange the bow warrior monk unit.

Matchlock units scale in skill in the order of ashigaru, samurai then monks. The warrior monk variant has a special ability that allows it a slightly longer range for a short amount of time. It's a good idea to position matchlock units on your flanks for a clear firing view and two massive morale shocks from a) being shot with matchlock rifles and b) being attacked from the flank. Matchlock units are also highly effective against cavalry.

Kisho units such as ninjas can remain hidden from enemy view while idle or walking on the battlefield. Ninjas have special abilities such as grenades which can cause devastating damage against all units.

Light cavalry are weak cavalry units that are really only good for scouting for hidden enemy units, wasting enemy ammo or sacrificing to a hostile charging cavalry unit.
To be honest it's good to have one or two to keep as sacrifices.

Katana cavalry are exactly that, katana cavalry. They aren't very good compared to other cavalry options.

Yari cavalry is arguably the best cavalry unit in SHOGUN 2. They can take out enemy cavalry with ease thanks to their yari spears, and have a higher than normal charge bonus, so you can charge into an enemy unit causing high damage, then quickly escape before spears arrive as reinforcements.

Bow cavalry are effective hit-and-run units that can over time cheekily rip apart enemy units. They have high ammo counts and suprisingly decent accuracy. Because they are mounted, they can escape spears and can even double up as a light cavalry unit.

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Sayoonara!