In King Crusher, achievements resemble missions. There are 3 for each level, and like in many games where you get three-stars per level completed completely, here, you can complete all the 3 achievements for that "3-star ranking". The rewards are constantly gems, which's great since they will help amount to your Incantations. I recommend constantly reading what the achievements are prior to you start the level, due to the fact that like that you will know what to watch on and what to do in order to complete them and earn more gems. Do not forget that if you pass a level with 3 achievements completed, then you should also watch an advertisement to double those gems got. It's not truly worth enjoying an advertisement to double the gems if you just complete 1 or 2 achievements, however hey, you do whatever you like, it's your game!