If you are looking to buy a used XBOX 360, then ask the seller if it has been banned from XBLA, (XBOX Live Arcade).  I am not aware of any way to re-enter an XBOX 360 into XBLA once it has been BANNED FOR LIFE.

Your XBOX 360 can be connected through the internet to XBOX Live Arcade.  From here you can buy downloadable games and play games, both from disks and from XBLA, with other people.  You can play with, or against, individuals or groups.  This can include live voice chat during gameplay.  Gaming is great but it's even better when you play with others.

One problem, and you'll need to watch out for this...

Sometimes individules abuse others in the on-line gaming community.  Sometimes this happens a lot and the XBOX 360 game system will get banned from XBLA.  The hardware gets banned so that the abusive individual can not create a new user name and continue abusing others.

Persons new to gaming may not have any interest in on-line play through the XBLA and may focus on playing the game disks in single player mode.  But as your gaming matures having the option to play and talk with others, in game, may be highly desirable.  Kids may share gardens with others in Viva Pinata and adults can trash talk their buddies while playing Halo 3 with a dozen other people they know.

Remember to ask if the Xbox 360  has been banned from XBLA live!  Let me know if this helped you!