One piece wheels are made of a single casting, while two piece wheels have a cast or forged center that is attached to the outer barrel, either by welding or via bolts. Three piece wheels have a center and two separate portions of the barrel attached to each other, almost always via bolts.
One piece wheels have advantages include simplicity, low cost, and lack of necessity of maintenance.
Two piece welded wheels are able to be constructed to a specified set of dimensions (varied offset / backspace).
Two piece bolted wheels consist of the face and the barrel, attached to each other via bolts. A major advantage of this type of wheel is that if the barrel is bent or otherwise damaged, it may be possible to be replaced without replacing the whole wheel.
Three piece wheels have two piece barrels and a face, and in some cases are able to be modified by changing one of the barrel pieces to make a wider (or narrower) wheel.