This is the easiest game mechanic to explain and probably one of my favourite ones as well. The achievement game mechanic is simply a mechanic where you earn achievements for accomplishing specific tasks. This is a screenshot from World of Warcraft achievements screen.
There are also other games and platforms that use the achievements game mechanic. For example, the Play Station network has its own trophies for games.
Even WordPress has its own badges that you earn from doing things. Like posting specific amounts of times etc.