This game mechanic is used to reward the player and commonly happens outside of the game. Commonly used to level up the player’s profile. Steam and Xbox use achievements for every game on their systems that level up the player’s profile showing other players the skill of that player.
Telltale Batman uses this mechanic as a way to show the player their progress into the game. When the player reaches a certain part of the game they will receive an achievement that appears on the screen.
DC Universe Online has an in-game achievement system, unlike most games that reward the character of that player with unique items. The achievements are gained for completing all different task such as completing certain parts of the story or for performing a combo such as defeat 5 enemies within 10 seconds.
In my game, I would make it so that the achievements affected the player in the game. If the player earned the achievement they would receive awards that benefited their character or vehicle.