Achievements in Fishdom are a chance to get diamonds for doing what you normally do anyway.
All the available in-game achievements are listed in the Achievements section. Tap the trophy icon in the left sidebar to see it. Every achievement comes with a short description of what you need to do in order to unlock it.
Almost all the achievements have three levels marked by trophies. Reaching new levels earns you a reward, so you get diamonds three times per achievement.