What do the Xbox 360 achievement points (or "gamerscore") do?

It's pretty much just for the bragging rights, you can't "cash in" your score.

But you can get freebees for using your Xbox. If you are an Xbox Live Gold member on Xbox Live in certain regions you are eligible for the rewards program, read all the details here: http://rewards.xbox.com/

Although it used to mean nothing but how "eleet" you are, the Xbox Live Rewards system has announced that you will get increasing benefits based on your gamerscore, including discounts and free gifts, with the highest tier unlocking at 25,000 gamerscore. There's nothing particularly exciting on offer, but signing up is free.

Beyond this, there are a few other things you can do with the Xbox gamerscore/achievement system:

• Getting all the achievements in a game (typically 1,000 gamerscore in a retail game, and 200 in a XBLA game, usually plus some if there's DLC) will add that game to your "Completed" list, which tracks all of your completed games.
• It's possible to select a game, look at the achievement list, and compare your score on that game to another person's. This can give you an idea about how far they've played in a game compared to you, or whether or not you both own a particular game. It will also allow you to join current friends who are playing the game, in a game party for that specific game.
• Microsoft publishes an API for interacting with the gamercard system, which people have used to create online playercard generators, which you can use to add your gamerscore and recently played games, among other things, to a forum signature or similar.

Gamerscore doesn't have any function other than bragging rights. Each account keeps a running total of Gamerscore earned across all games using that account.