You're normally not that unhelpful.
Off to Google and Wikipedia, then.
Cheers!
Sorry. It's really hard to explain. Basically, Lebesgue integration is far more general than Riemann integration. In Riemann, you are always approximating areas with rectangles. (Bottom line: that's pretty much all Riemann does.) Lebesgue uses measure spaces (sigma-algebras, if you know the term). It's very hard and I haven't dealt with it in years. I could refresh myself if need be.
This is FreeRepublic. I didn't realize you actually wanted an answer!