If you are talking about ones like NSF, NDSEG, or Hertz, they cover your tuition and stipend so you don't need to get an RA or teach (NSF actually limits the amount you can teach since they want you to be doing research). If your stipend is less than the NSF, I believe you just keep the extra money. If your stipend is more than the NSF (mine is by a few hundred dollars) the school will often just supplement it.
However, at some institutions if you teach while on the NSF (which you can only do at most like 10-15 hours a week one semester per year) you will get paid to teach on top of your stipend.