Achievements are "special missions" that you are required to carry out as you play. A detailed list of these missions can be found in your profile. Just click on the Profile icon and scroll down to the Achievements session.
For each achievement you unlock, you get Points that help you proceed to higher Levels and unlock new Modes of the game!
When you achieve a mission you are not required to do something. The corresponding area in your profile is updated automatically and you collect the points.