A Gamertag is the terminology for user names over Xbox Live. You choose your gamertag once you sign up for the service and it is your "identity" in the Live world from then on. If ever you discontinue Xbox Live you will lose your gamertag and can never get it back. No one else can use the name either, so keep this in mind when canceling your account.
Vulgar gamertags can be reported to Microsoft and they can, and will, force you to change it. All Live-enabled games have the ability to report "bad" gamertags.
At one point, gamers were being forced to change gamertags that contained spaces. For example, if you had the gamertag "My Gamertag", you would have been required to select an entirely different gamertag. Those that applied were contacted via email and given something free for their inconvenience.
(My memory is a bit hazy on the space issue)
In the next iteration of Live to come out when the Xbox 360 hits the streets there will be a new concept called Gamer Card that is "attached gamertags". This will add new functions and labels to your online identity.