First, the background: All Xbox 360 games have Achievements, which are just milestones in the game. Some you can get just by playing; others require you to do something special to get them. Each Achievement is worth a certain number of points. Retail games have at least 1000 points, which Xbox Live Arcade games have 200. Extra downloadable content can add additional Achievements for additional points. Gamerscore points don't mean anything, other just for bragging rights.
One of the reasons I decided to break down and buy a 360 was an role-playing game called Mass Effect. After 5 playthroughs with 3 different characters, I finally earned all the Achievements (including the 50 points for the downloadable side mission).
I have 2 more games that I think I can get all the Achievements for. (Many games require online play to get all the Achievements. Since I don't play online, I know I can't get those.)