I see another poster stated you can download games. I'm not aware that that's a possibility. You can download additional content for games you currently own though. If patches and such are released for a game, normally you can download that free of charge. When new maps are added however, downloads can run roughly $5.00 per download. You have to click to download it though, so you choose if you wish to make that purchase. What happens is, the developers keep working on games that have already been released and they add additional levels and such. It just works out well because for a small additional fee, you get much more benefit from the game and the developers make a little more money, so they've got incentive to keep working on the title too.
One thing I will suggest however is, make sure when you sign up to live that you include your email address. Every time a download is made, MS will email you and tell you that $5.00 (or whatever amount) will be charged to your credit card. This allows you to keep your kids in check. We've purchased 3 Xbox's (old ones, not 360) and my son has downloaded this additional content on all 3 consoles.

I get SO darn angry at him!!! We have roughly 60 Xbox games and while they're not all Live titles, several of them are, and it adds up!!!!
We don't have the 360 yet, but I read something about it working on a point system. Perhaps you buy the points and then the kids can use those points for their downloads? I'm not too sure about that. Maybe it was just something MS came up with for users who don't wish to use or don't have a credit card. When you're talking lots of younger people, many fall into that bracket.
Either way though, you need to set the rules up with your children. Maybe make it a rule that before they download anything, they must inform you. Just remember that many of the downloads are free. If you get the email that they've downloaded without your permission, perhaps at that time, further disciplinary action is in order? IMO, you do need to set up some sort of system.