Although Xbox 360 doesn't guarantee a 100% compatibility on EVERY first-generation xbox game, there are some game (born for the original xbox) you can successfully play on the xbox 360
Here's a backward compatibility list:
http://www.xbox.com/en-US/games/backwardcompatibilitygameslist.htm
Being halo 1 and 2 major success for that console, and being them produced directly by Microsoft, it was almost certain for them to be worked out as "compatible" games.
One side note: halo 1 will be released on the xbox live marketplace with the next xbox update (scheduled for december 4th). Instead of buying a physical disk (although the choice is sometimes preferrable) you could download directly to your hard disk, if you have one.
Hope it helped