Achievements are awarded for completing certain milestones in a game. 
Once you click on a certain game under the “Games” tab of the Game Center application, you will be taken to the leaderboards(click for more info). Right next to it you can find the “Achievements” tab.
When you click on it, you will be able to view the complete list of achievements that you can earn in the game, as well as the achievements that you’ve already got.

Achievements are awarded for completing certain milestones in a game. They are different in every game, and while some achievements can be pretty easy to earn, others will require a lot of attempts. For each achievement that you earn, you will be awarded points depending on the difficulty.

Under the “Achievements” tab you can also create new challenges for yourself and your friends.
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