Each of the 12 stages have 10 awards, so in order to receive this achievement you must get every one. Stages 1 through 10 share similar designs to their awards, which will require one of the following conditions:
Beat the stage - Obtained simply by defeating the boss monster, and progressing the storyline.
Remove a certain barrier - These are specific wooden barriers located around each stage map. They can be removed by stepping on the corresponding trigger tile (the isometric square panels). You will need to remove all of them in each level to get around regardless, so it shouldn't be an issue getting the appropriate award barrier. Some may not be accessible until you have completed the stage.
Defeat a certain monster - These can be distinguished by an icon by the monster's name when encountering them on the dungeon map. They will typically stand out, by either being stationary in dead ends, trapped behind wooden walls, and so forth.
Defeat a lair of monsters - When you walk onto a lair on the dungeon map, you will fight a boss monster, a stronger version of the beasts produced by the lair. Defeating the boss monster will destroy the lair, giving the appropriate award. Not all lairs are connected to an award, though you will most likely want to destroy them all regardless so shouldn't have any issues finding the award lairs.
Open a certain treasure chest - This is the rarest of the standard award criteria. Some chests in the Grove of Prey stage will be destroyed by monsters if they reach them before you. You will have to go through the level once or twice to open up all the barriers before you will be able to have a path through to the chests. Then it is just a matter of racing to the appropriate chest.
Score 1,000, 3,000, and 5,000 points - Simply score the appropriate value. This is extremely easy to farm from a lair, as they will constantly spawn new monsters. While each lair can only spawn a limited number, there is still far more than enough.
In order to receive easy points each fight, use a monster such as the 2nd and 3rd tier Beetles (with Trident). Damage the opponent to low life, then let your ability gems build up. Make it a priority to remove the enemy ability gems so they don't destroy your own. Simply keep building up until it is impractical to continue, and then match a Trident. Trident will destroy an extremely large area, destroying enemy ability gems, rocks, as well as your other ability gems--setting off a chain reaction of more Tridents, covering most of the field. You should receive a very large overkill bonus (several hundred, I've seen as much as a thousand) as well as receive a large number of puzzle points simply from the build up period and any cascades or rocks/enemy gems destroyed at the end.
Stages 11 and 12 have very different Awards than the other 10. Stage 11 will initially show "???" for the Awards until beaten for the first time (ie: complete the game). Afterward you will be tasked with defeating the leader of each affinity type, each being an award, as well as an extra "secret boss". You will need to take the teleporter to the sides of the level(only available after completing the stage and going back) to open the barriers. Then in the central area you fought Everett (the final boss) you will see paths branching out to a number of lairs. Each lair corresponds to a different affinity/beast and award. After clearing all of the affinity lairs and their bosses a new final enemy will spawn where you originally fought Everett, defeating him will give you the final award for the stage. (Note: You must clear all the lairs in one go to spawn the boss. If you die or leave the level you will have to start over, though the barriers stay down.)
Stage 12 is the Challenger's Door, and the awards here are where the going gets really rough. It is often easiest to replay to first fight rather than risk attempting them in a real push through the stage. You don't want to be near the end and get hung up trying to get Awards.
While some information is reproduced here, Raven2Go's forum post on the Challenger's Door is highly recommended reading. (Link here)
Challenger's Door - Complete the stage Challenger's Door. This one is obvious enough. There are 15 battles total, fought survival style (without hit points restored between fights). Forum member Raven2Go recommends using Goliath (3rd tier Spider), as his ability to destroy locks for damage can get you through situations that otherwise have very difficult to avoid damage. See Raven2Go's post linked above for more details on the battles and some great tips on the boss.
Overkill - Win a battle with a killing blow that deals more than 10% of your foe's max HP. Again, an obvious one. You should be able to get this right away, just make sure you use a high damage monster.
Not a Scratch - Win a battle with your beast at max HP. This will have to be done right out the gate, simply don't get hit during the first battle.
Close Shave - Win a battle while your beast has less than 1% of its max HP. While you can certainly get this via shear luck simply attempting to defeat the Challenger's Door, most likely you will consistently end up just above 1% life if not killed outright. For obvious reasons you will want to keep attempts at this award to the first few battles, restarting at the start and doing battle with the Cait Sith and other early monsters. Using a high HP monster is also very important obviously, so go with either the Gigas at 4200 or one of the ones that weigh in around 3000. 1% would of course be your beast's life with the last two zeroes removed. (I feel stupid for even feeling the need to say that.)
Rise to the Challenge - Win a battle using a beast that has less than 20% of its max HP at the start of the battle. This will probably come naturally as you play through. If you need to try specifically for it, just take some damage while setup to finish your opponent off and wait until you get below 20%. Finish off your opponent, and then take care of the following one as well.
Waste Not, Want Not - Win a battle without making a single idle twist (a twist that makes no sets). Again, an obvious one. You shouldn't have any trouble here, though it helps to use a high damage beast and destroy your opponent before you run out of matches.
Blank Slate - Destroy all the gems on the board with a single twist. This can be irritating due to shear bad luck, though the method is simple enough. Use a Phalanx (3rd tier Beetle), and let Tridents build up on the board. Wait until they are positioned so that the width of their blast will take out every column. (Idle twist them into place if you have to, but be careful they arent destroyed or not lined up properly to set one another off.)
Cascade x 7 - Make a seven cascade with a single twist.
For the most part this is simple luck, though having an abundance of gem changing ability gems out (such as the Dark Hunter's (3rd tier Ant) Nightmare ability) will aid in triggering more cascades. (Raven2go recommends the Colossal Watcher.)
7- Gem Match - Make a match of seven or more gems in the same row with a single twist.
This is easily the most difficult of the game's Awards. Outside of shear dumb luck(which you most certainly shouldn't bank on in the slightest) you will have to repeatedly fight the first Caith Sith and try to setup one of the 7 match designs detailed under "The Knowing" achievement. This can take many attempts, so don't go in expecting to knock it out the first few times. You have to be very careful about enemy gems disrupting the setup. While removing them is of course your first priority, if you can't you should twist them away so as to keep the explosions out of reach of the gem setup. You should definitely not use a beast that is lock or rock oriented, and using something like the Sol Volk that replaces enemy gems can also help.
The Perfect Twist - Make 4 or more sets with a single twist. The wording is just a strange way of saying make a Combo x4. Reference the diagram for Combo Stones under "The Knowing" achievement. This might seem daunting while battling for your life, but you will most likely get it during normal play by chance. Just keep an eye out for parts of the field that have multiple sets of two which would probably be a large focus anyway. If you need to you can sacrifice a game with idle twists to setup the combo if there is a spot that is close but not quite there.