A belt clip is designed to do just what it sounds like: Clip onto your belt. It's a bent piece of spring steel designed to fit snugly onto the edge of a belt or fabric; much like the clip on a pen cap.
An alligator clip consists of two, serrated spring loaded jaws that can be opened by squeezing them and are held shut by the spring. Alligator clips work best to attach a badge reel to a lanyard or thinner fabric that may not be thick enough to retain a belt clip.