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).
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, do not even provide gamerscore based on how difficult an achievement is. almost all of the game's achievements and gamerscore are based on multiplayer, but the campaign achievements are much harder to get and not worth as much.
The worst of all, glitched achievements are achievements that obviously were not tested before release of a game, and were not programmed right, so even if you do what the achievement requires, it will not unlock. There is a Ghostbusters game and a Tiger Woods game that have multiple glitched achievements
The best are stackable achievements. They make it so you can get all of the achievements by playing on the hardest difficulty rather than beating a game on every difficulty (example: GH3 did not have stackable achievements).
Online achievements are both good and annoying. They are good because they encourage people to play a game on xbl if they have it. They're bad because it requires money to get xbl, so really it's all probably just a marketing ploy to get people to buy xbl.
Co-op achievements are the most pointless and stupid, since it's usually easier to beat a game with more than one person (Example: Saint's Row 2). There are other games where it requires both people to be good at the game (Guitar Hero games), but why should two people have to be good at a game just to give one person an achievement? If that one person is the greatest person at the game, don't they deserve all the achievements?
There are also virus-style achievements, which basically start off with one person, then they unlock for anyone who plays with them. Works for up to 10 generations. These achievements take no skill and usually aren't worth gamerscore.
There are also bad achievements, which you get for doing poorly. They are either wort no points, or, unfortunately, usually are worth points (Such as the "Fail a song" achievements in GH games). I don't really understand it. They give you points just for proving you don't have a massive ego.
There are also DLC achievements, which add on to the original game's achievement list and gamerscore, allowing more than 1000 total gamerscore. Sometimes, the company who made a game will actually be able to add more non-DLC achievements (Examples: Halo 3 and Fable 2), which are added to the achievement list via an online patch. Most games just stick to 1000 gamerscore however, and some DLC does not offer extra achievements, so be warned.
Some games, however this is rare, have achievements for unlocking all the other achievements in the game. F.E.A.R. 2 has an achievement for unlocking all multiplayer achievements. It's just to provide a larger gamerscore gap between the people who are missing a few achievements and the people who have them all.
Last, but not least, Missable achievements are achievements that can be gotten in a single playthrough of a game, but can be missed if you're not careful. There's a lot of these in a lot of games, so it is advised that you use achievement guides before playing a game.
The PS3 equivalent of achievements are "trophies". Bronze for easy tasks, Silver for medium tasks, and Gold for challenging tasks. There are also Platinum trophies for getting all trophies in a game. This was the PS3's attempt at copying a popular Xbox 360 feature (Much like when Microsoft copied the Wii's "avatar" concept, and treated it as if it was their own idea)
Achievements are a neat concept, but could have been thought out better.
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."