Yes, about 300+ X-Box games work for the 360. However, you can only play them on the Elite, or you can get a software emulator for the original 360/360 Arcade.
Go here for the list of compatible X-Box games:
http://www.xbox.com/en-US/games/backwardcompatibilitygameslist.htm
UPDATE:You cannot play Xbox LIVE on Original Xbox games. :(
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-MidKnight777, DevinUser514
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No, only Xbox 360 games work on Xbox 360.
No, but some original Xbox games work for Xbox 360.
Not all xbox games work on the 360. But heres a website with a list of xbox games that work
No. NTSC games will not work on a PAL Xbox 360.
No they do not, but you can play certain Xbox original games on Xbox 360