Many types... The basic categories would be open or closed, i.E whether there is an associated break in the overlying skin exposing the fracture to the outside and thereby increasing chance of infection. Also, intra or extra articular, i.e.Whether it is involving the joint. Also, comminuted, i.E whether the break involves two or more pieces. And lastly, displaced, that is whether the pieces have moved or not.
A LOT . We have 206 bones. And classification depends upon a variety of systems. There are "named" fractures that frequently use the name of the describer ( ex abraham colles described the "colles fracture" of the radius) but many of these are not accurate. We use lets say n terms such as nondisplaced/displaced angulated/nonangulated comminuted simple transverse butterfly etc/ multiply by 206(n!) a lot.