The XBOX 360 is a powerful system with titles for hardcore gamers. Online play is critical to having the real 360 experience as it allows you to compete globally and chat with fellow players to create more of a social aspect to what was once a anti-social activity.
Genre wise, it seems like the games that shine for XBOX 360 tend to be related to sports or war. You've got recent popular titles spanning different genres like Lost Odyssey, Fable 2, and Street Fighter 4 as well.
The XBOX comes in 3 versions, some with more GBs than the others. The GBs are for downloadable content - previews, games, and other things you want to save on your system.
One of the major problems with the 360 that most people I know encounter is the RRoD - red ring of death. This system, although made with a cooling fan, tends to overheat so you HAVE TO take breaks often while playing otherwise you'll end up stuck paying repair costs. A lot of times people just end up having scares like the system won't turn on or red lights appear but it can be easily resolved. Once I just had to unplug everything then put it back in and it was fine.
I think if you're a newcomer to gaming or don't really like gorey games, the 360 may not be for you. It seems geared toward more teenage or adult males, although there is a small number of hardcore girl gamers slowly materializing.
So overall - powerful system, you gotta pay for online play but its really amazing stuff (chatting, emailing, etc with others) and there's a growing variety but at the same time, for the price and risk with this system, you may be skeptical about it and opt for Wii or PS3 which have free online play and don't seem to carry the risk of breaking down when you play them for extended periods of time.