On the Xbox 360, achievements are basically bragging rights. Once an achievement is unlocked, by doing whatever it says in the achievement description, you get the gamerscore the achievement was worth. Generally, normal 360 games can have up to 50 achievements worth 1000 gamerscore total. Arcade games usually have up to 12 achievements worth a total of 200 gamerscore. There are a few rare cases, like Quake 2 (which has 9 achievements, worth 0 points).more...
It is frequently complained that sports games and kids games are too easy to get 1000 gamerscore, and lots of games have basically got "giveaway" achievements that take no skill to get. And some harder games, like Ninja Gaiden 2 are nearly impossible to get a decent amount of gamerscore on. Also, secret achievements can be a big hassle, since there is no description, so you either have to look online, or guess how to unlock them. Other achievements, such as the achievements on Perfect Dark Zero
Awards you get for fulfilling certain tasks in Xbox 360 games. They vary in score, usually more difficult achievements being higher score than less difficult ones. Most retail games have 50 achievements worth 1000 gamerpoints and arcade games have 10-12 achievements worth 200 gamerpoints, although there are exceptions (Halo 3, Gears of War 2, etc). All these achievements add up to your gamerscore, which has become something to brag about with many people.
There are also people who simply buy or rent games just for the gamerscore. They are known as achievement junkies or achievement whores. I, being a reformed achievement junkie, don't really see the point in getting games just for their gamerscore. You should get games because you like them, not for the achievements.
Person 1: "Bob has been in his house for three straight days, what's he doing in there?"
Person 2: "He just got a new Xbox 360 game and he has to get all the achievements for it. He's an achievement whore."
Person 1: "I see."