Destroy runes through matches by matching them with runes of the same type with 3 or more in a row. (runes destroyed by spells or cascades do not count, probably not bombs either)

Air Runes

Air Apprentice

You're getting good at that air thing. You've made 10 air matches!
Destroy 10 air runes through matches.

Air Novice

Congratulations! You've gained novice proficiency with air.
Destroy 100 air runes through matches.

Air Journeyman

You've made it pretty far down the journey on the path of air.
Destroy 500 air runes through matches.

Air Master

Masters of the Universe? Nope, just of air.
Destroy 1000 air runes through matches.

Air Savant

Savant, now that's a nifty word, and you totally deserve it in all matters regarding Air.
Destroy 5000 air runes through matches.

Fire Runes

Fire Apprentice

You're getting good at that fire thing. You've made 10 fire matches!
Destroy 10 fire runes through matches.

Fire Novice

Congratulations! You've gained novice proficiency with fire.
Destroy 100 fire runes through matches.

Fire Journeyman

You've made it pretty far down the journey on the path of fire.
Destroy 500 fire runes through matches.

Fire Master

Masters of the Universe? Nope, just of fire.
Destroy 1000 fire runes through matches.

Fire Savant

Savant, now that's a nifty word, and you totally deserve it in all matters regarding Fire.
Destroy 5000 fire runes through matches.

Water Runes

Water Apprentice

You're getting good at that water thing. You've made 10 water matches!
Destroy 10 water runes through matches.

Water Novice

Congratulations! You've gained novice proficiency with water.
Destroy 100 water runes through matches.

Water Journeyman

You've made it pretty far down the journey on the path of water.
Destroy 500 water runes through matches.

Water Master

Masters of the Universe? Nope, just of water.
Destroy 1000 water runes through matches.

Water Savant

Savant, now that's a nifty word, and you totally deserve it in all matters regarding Water.
Destroy 5000 water runes through matches.

Earth Runes

Earth Apprentice

You're getting good at that earth thing. You've made 10 earth matches!
Destroy 10 earth runes through matches.

Earth Novice

Congratulations! You've gained novice proficiency with earth.
Destroy 100 earth runes through matches.

Earth Journeyman

You've made it pretty far down the journey on the path of earth.
Destroy 500 earth runes through matches.

Earth Master

Masters of the Universe? Nope, just of earth.
Destroy 1000 earth runes through matches.

Earth Savant

Savant, now that's a nifty word, and you totally deserve it in all matters regarding Earth.
Destroy 5000 earth runes through matches.

Void Runes

Void Apprentice

Should you be dabbling in that type of magic? You've made 10 void matches!
Destroy 10 void runes through matches.

Void Novice

Dark magic isn't for novices, so be careful!
Destroy 100 void runes through matches.

Void Journeyman

Avoid the Void! You don't want to keep going down this path.
Destroy 500 void runes through matches.

Void Master

You're now a master of the dark arts. Stupifexpelierminous!
Destroy 1000 void runes through matches.

Void Savant

Wow. You might have to uninstall before your darkness eats your computer.
Destroy 5000 void runes through matches.

All Runes

Rune Master

So many runes, so little time. You've truly mastered the the manipulation of runes.
Destroy 50,000 runes total across all types through matches.

Rune Savant

You're such a savant that there are no more words to describe your awesomeness.
Destroy 100,000 runes total across all types through matches.

Triads

You get a triad rune from matching 4 runes and a gem rune by matching 5 runes in a line.
Gem runes are worth about twice as much as triad runes.
Gold and chests can drop from enemies when they are killed, and disappear after a while if not collected in time.
Gold piles are worth about twice as much as gem runes, while chests are worth about twice as much as gold piles.
So if you are able to take your eyes off the runes often enough to click the gold piles and chests, these achievements will take less time.
In addition, the value of all 4 increases at higher levels.

The total number of triads collected from the battlefield is what counts towards these achievements.

Gold Digger

Digging up triads is your forte. You have collected 1000 triad runes on your board.
Collect a total of 500 triad runes (the in-game counter increases by 2).

Thrifty

Save those triads for a rainy day. You'll need them.
Collect a total of 100,000 triads from the battlefield.

Triad Pincher

A triad saved is a triad earned.
Collect a total of 250,000 triads from the battlefield.

Miser

You need to go swim in your triads today.
Collect a total of 500,000 triads from the battlefield.

Health Runes

Matching 5 runes in an L or T shape makes a health rune.

Team Medic

You don't need anyone else to heal you. You have used 1000 health runes.
Use 1000 health runes.

Bomb Runes

Matching 6 runes makes a bomb rune.

Bombastic

Bombs are your specialty.
Use 1000 bomb runes. (this one takes a while)
To make it a bit easier/faster, try using the water swap spell to turn a potential 5 match into 6. Doing this on the bigger maps, and/or the maps with only 3 rune types, also helps.

Big Bomb Runes

Matching 7 runes makes a big bomb rune.

Fire Everything!

Explosions never felt this good.
Use a big bomb rune.

Big Bada Boom

Your explosions are so big that they can probably be seen from space.
Use 10 big bomb runes.

Variant (bad) Runes

Whole Lot of Nothing

Corrupted, schmorrupted. You'll break any runes you can find.
Make a number of matches that include a corrupted rune.

Ice Breaker

Nothing stays frozen around you, you ice breaker!
Make a number of matches that include a frozen rune. (number not shown)

Policy Themes

These are unlocked by having a majority of your policies favoring a certain theme by the time you enact your 35th policy (the policy choice you get after beating level 35.) This means you will have to play through level 35 on at least 4 different cities. (recommend doing this on easy difficulty on as many cities as possible, though you will need cities at normal and hard difficulties for other achievements, so keep that in mind)

The Mystical Ruler

Magic permeates the very fiber of your town.
Choose policies that favor the tower of elements, or anything that involves the tower of elements or magic.

The Republic

Power to the people! Well, at least they can vote on it.
Choose policies that let people vote, or establish committees to make decisions.

The General

This is madness! No, this is ReignMaker!
Choose policies that favor the military, or anything to do with warriors.

Animal Planet

Peace and love to all who live in your town. Oh, and a few human sacrifices.
Choose policies that favor the druids, or protect nature.

City Building

Apprentice Builder

This is the house that you built.
Upgrade a building to level 2 (and maybe all other buildings to level 1).

Journeyman Builder

Things are starting to look nice around here.
Upgrade all buildings to level 2 (and maybe at least one building to level 3).

Expert Builder

This is getting to be a rather splendid place.
Upgrade a building to level 4 (and/or maybe have all other buildings to level 3).

Master Builder

Your city is looking mighty snazzy!
Upgrade every building in a city to level 5 (the max level).
This will take a long time, and a lot of resources, especially wood, so make sure you upgrade your lumbermill first, and check back into the city regularly to collect the resources so they can keep producing.

Blacksmith Building

Tinker

You've definitely perked that smithy up.
Upgrade the blacksmith to level 2.

Master Crafter

No finer place to craft has ever been built.
Upgrade blacksmith to level 5.

City Workers

Farmers Union

Have a the maximum number of farmers allowed. (hidden)
Hire 20 farmers in your town hall.

Spending Triads

Little Spender

You finally have some buying power. You've spent 5000 triads.
Spend 5000 triads.

Middle Spender

Buy, baby, buy! You've spnt 50,000 triads.
Spend 50,000 triads.

Big Spenda'

It's only money, right? You've spent 100,000 triads.
Spend 100,000 triads.

"Reading Minds"

You can check what your citizens are saying by clicking on the speech bubble above their heads

Mind Reader

Let's just take a peek at those thoughts.
Read the mind of a citizen.

Psion

You must be psychic!
Read the minds of 10 citizens.

Master Psion

Your citizens know you spy on them with care.
Read the minds of 50 citizens.

Levels

Hegemon

You've just crossed beyond the Threshold of greatness by beating 20 levels!
Beat level 20 on any difficulty.

Easy Peasy

That wasn't so bad, was it? Good job defending the towers.
Beat level 50 on easy difficulty.

Middle Man

Excellent work defending the towers. Here's a lousy award for it.
Beat level 50 on normal difficulty.

Master of the Tower

We bow to you, the Master of the Tower.
Beat level 50 on hard difficulty.

Master Beater of the Tower

You are the master of beating towers. (hidden)
Beat level 10 of The InfinitY Tower (level 51.)

Master Beater Forever

You are an even better master of beating towers. (hidden)
Beat level 25 of The Infinity Tower (level 51.)

Bosses

It is possible to get the achievement for killing a boss without beating the level if you die before killing all the minions, but you need to beat the level to progress.

Apprentice Becomes the Master

You beat the Tutor in the tutorial. Congratulations!
Beat the Tutor "boss" in level 1 on any difficulty.

What a Prick

You beat the Godgantupine. Congratulations!
Beat the Godgantupine boss in level 5 on any difficulty.

Together at Last

You beat Thargorina. Congratulations!
Beat Thargorina (the boss) in level 10 on any difficulty.

Void the Defeat

You defeated the Obliterator, a servant of the Void. Congratulations!
Beat the Obliterator boss in level 15 on any difficulty.

Not in Kansas Anymore

You beat the Wizard whose face is strangely familiar. Congratulations!
Beat the Wizard boss in level 20 on any difficulty.

Flung The Flinger

You beat the Elemental Lord, something the Alchemists claimed they did. Congratulations!
Beat the Elemental Lord boss in level 25 on any difficulty.

Got My Eye on You

You beat that rolling eyeball known as the Glaring Glancer. Congratulations!
Beat the Glaring Glancer boss in level 30 on any difficulty.

Got Boned

You beat Bizarre Bones. Congratulations!
Beat Bizarre Bones (the boss) in level 35 on any difficulty.

So. Many. Pokka.

You beat the Cubicle. Congratulations!
Beat the Cubicle boss in level 40 on any difficulty.

Unchained Melodrama

You beat the Obliterati. Congratulations!
Beat the Obliterati boss in level 45 on any difficulty.

Heal OP

You beat General Mabelrode and saved the world. Congratulations!
Beat General Mabelrode (the final boss) in level 50 on any difficulty.

Monster Mastery

Turtoom Mastery

You did a good thing in destroying these poor, corrupted creatures.
Kill 1000 Turtooms of any variation.

Turtoom Boom Boom

We shall mourn these good turtooms.
Kill 2500 Turtooms of any variation. (this one will take a while)

D'orc Mastery

Good job in destroying the d'orcs!
Kill 1000 d'Orcs of any variation.

Gobbo Mastery

The gobbo horde has been contained.
Kill 1000 Gobbos of any variation.

Strange Creatures Mastery

Strange creatures? Cute but no cigar!
Kill 1000 Gelatinous cubes.

Gargantupine Mastery

The gargantupine threat has been contained.
Kill 1000 Gargantupines of any variation.

Void Mastery

Avoid the Void. That's definitely a new saying.
Kill 1000 Void enemies of any variation.

Dragonet Mastery

Excellent work in dispersing the crazed dragonets.
Kill 1000 Dragonets of any variation.

Romel Mastery

Horrid creatures! All romels need to be destroyed.
Kill 1000 Romels of any variation.

Evil Slemor Mastery

What kind of horrible beings would corrupt slemors?
Kill 1000 Slemors of any variation.

Undead Mastery

Who doesn't like fighting the undead?
Kill 1000 Skeletons of any variation.

Glancer Mastery

Fighting eyeballs? At first you were the pupil, but now you are the master.
Kill 1000 Glancers of any variation.

Perfection

Gold That Glitters

Gold flags for every tower? Maybe YOU should be the king!
Get gold on every level with a single city.
To get gold you need to beat the level with at least 95% of your tower health remaining.
Level 51 (The Infinity Tower) counts as gold if you get gold on any of the levels there.
Doing this on easy difficulty is recommended.
For the bosses you may need to bring strong spells fully charged, as well as the healing spell and possibly also healing troops, so you can heal up after the boss is dead before killing its last minions if needed.
If you are having trouble with getting gold on a boss, one (expensive) solution is to have the armory at level 5 and stocked up on direct damage items, and dumping them all on the boss at the start.

Blacksmith

You gain some of these achievements by researching the gear recipes in the armory, but for the later ones it seems to sometimes be enough to complete the levels where the recipes are found as loot at the end of the level, without actually needing to research them.

Apprentice Blacksmith

Congratulations! You learned how to make something.
Research 1 recipe.

Journeyman Blacksmith

Congratulations! You learned to make a few things.
Research 5 recipes.

Expert Blacksmith

Wow, you sure do know how to make a lot of things!
Research 10 recipes.

Master Blacksmith

You are a master of metal, indeed. You know all the blacksmithing recipes.
Research all 12 recipes.

Bestiary cards

This set of achievements is earned through filling the bestiary with cards you discover by collecting the scrolls that sometimes drop from killed enemies.
To unlock all the achievements for filling the Bestiary with these cards, 45 different cards are needed in total. This means getting the scroll from all the different normal enemies and bosses.

The drop rates can seem quite low, especially for the bosses, so you will probably have to defeat some of them a lot of times.

There was a rumored 46th scroll from the Pokka minions of the Cubicle (Gelatinous Cube) boss, but that was a bug and has been fixed, and it did not count toward the achievement anyway.

The enemy that is usually the hardest to find is the Shontan Gargantupine, but since The Infinity Tower (level 51 on the map) randomly cycles through all the different enemies, this seems to be the most reliable place to find it.

See the Completed Bestiary section at the bottom of the guide for images of a completed bestiary.

Dog Catcher

Discovered your first bestiary card. Gotta get them all!
Discover 1 bestiary card.

Zookeeper

You're starting a rather nice collection there.
Discover 10 bestiary cards.

Monster Lover

Beasts! Gobbos! D'orcs! Oh my!
Discover 25 bestiary cards.

Gotta Catch Them All

I choose you and your little dog, too! You've collected 40 bestiary scrolls.
Discover 40 bestiary cards.

Beast Master

Who knew there were so many creatures? You've got them all! (hidden)
Discover all 45 bestiary cards.

Achievements

These achievements are for getting the required number of achievements in a single city/playthrough.

Ding!

You unlocked your first (and now your second) achievement.
Unlock 1 achievement.

Ding, Grats!

You've achieved 10 achievements. What an achievement!
Unlock 10 achievements.

Gifted

You've achieved 25 achievements. What an achievement!
Unlock 25 achievements.

Very Gifted

You've achieved 50 achievements. What an achievement!
Unlock 50 achievements.

Achiever

You've achieved 100 achievements. What an achievement!
Unlock 100 achievements.

Over-Achiever

Congratulations! You just earned an achievement for earning all the achievements. (hidden)
Unlock all the achievements possible to earn in a single city/playthrough, as shown in the achievements menu in game.

The L Award

It's okay that you lost. It's not like the fate of the world rests on your shoulders.
Lose once.

Frobs It All

This losing thing, it's getting to be a habit!
Lose 5 times. (the in-game counter increases by 2)

The Frobinator

If you can't say something nice, say nothing at all. (You lost again.)
Lose 13 times. (the in-game counter increases by 2)

QQ

How did you manage to lose to the Tutor? That's impossible! (hidden)
Lose to the Tutor on the first level. This takes a while, you just have to resist the urge to kill him while he whittles down the health of your tower.

Anxiety Attack

Calm down and take a deep breath! You made three bad moves in a row. (hidden)
Make 3 bad moves in a row.

Healing

Hire healers and high priests from your town hall, and use them in battle so they can periodically heal your tower. The tower does not have to be damaged for the healing to count. A single healer can keep healing through a whole level if you keep them alive, but if the tower takes too much damage they will die, and you'll have to use another one, or do it again from the beginning of the level next time.

Apprentice Healer

You've healed your tower.
Let your healer(s) heal your tower for 1000 health points.

Journeyman Healer

Who knew towers needed so much healing?
Let your healer(s) heal your tower for 5000 health points.

Master Healer

If anyone needs a tower healed, they should come to you, the Master Healer.
Let your healer(s) heal your tower for 10,000 health points.

Troops

Hire troops from your town hall. Higher town hall level gives access to better troops.

Heals, Please!

You hired a healer from your people.
Hire a healer from your town hall.

Healers, Healers Everywhere

You hired some healers to keep things healthy.
Hire 5 healers from your town hall.

Heal Me, We Win!

Heals for everyone! You've hired a high priest.
Hire a high priest from your town hall.

Drood

Hired a druid from your citizens
Hire a druid from your town hall.

Droods

Holy fluffy bunnies - you got yourself a lot of druids.
Hire 5 druids from your town hall.

Many Droods

Back to nature! You hired an archdruid.
Hire an archdruid from your town hall.

Stone Roller

You got yourself a stoner, man.
Hire a stoner from your town hall.

Rock and Hard Place

You hired a bunch of stoners, and the rocks rained down.
Hire 5 stone rollers from your town hall.

Hard Rock Hero

Whoaaaah. That's a heaaaavy rock.
Hire a mega stoner from your town hall.

Point and Click

You hired an archer.
Hire an archer from your town hall.

Shoot 'Em Up

You hired some archers, and they're very into pin cushions.
Hire 5 archers from your town hall.

Pew! Pew!

You hired not just any archer, but an elite archer!
Hire an elite archer from your town hall.

Peek-a-boo

Hired a scout from your citizens.
Hire a scout from your town hall.

Hide and Seek

Hired a bunch of scouts from your citizens and sent them to the battlefield.
Hire 5 scouts from your town hall.

Search and Destroy

Hired a lead scout from your citizens, and watch him scurry around the field.
Hire a lead scout from your town hall.

Watch 'em Go

Your scouts surely didn't sign up for this! (hidden)
Have your scouts pick up 10 items from the battlefield.

A completed Bestiary will look as follows:

If you see any errors or know something I don't, please share it in a comment!