As I understand it, not playing any games that use them, 'achievements' can range from being sort of like merit badges that you earn, to gold stars you get for trivial tasks. There are a variety of achievements (kill a thousand zombies, make it through a tough area avoiding fights or only using weak weapons, complete a set of items, explore an area or find hidden locations, etc.). The reward can be something such as a title (being referred to as '{name} the orc slayer' if you wipe out an orc encampment) or a listing in the game stats (similar to DD's traits page), or they can unlock equipment, quests or certain bonus areas in a game.
After a quick google search, I've learned that the Ghostbusters game has an
achievement for knocking yourself down with your own weapon. World of Warcraft
has one for killing 15 turkeys in 3 minutes, and another for using a bandage to heal a character with less than 5% of their hit points left.
