Rabbit skin glue has a few drawbacks, but its big advantage is retaining flexibility once set - so it's popular for artists or work with fabrics. It's also the ingredient in gesso for sculptors or gilders.
"Hide" or "Scotch" glue is made from the trimmed scraps of hides but mainly cartilage or tendon. It's not made from hooves or horns - it could be, but they'd take far too long to render down. It used to be made as bricks or "biscuits" than needed to be broken up before soaking and heating, but these days it's all "pearl" glue that is quicker to soak.
Hooves were granulated and went for either case-hardening compounds (before the cyanide processes) or as garden fertiliser.
Fish glue is made from the cartilage of non-boney fish (sharks or dogfish) and sometimes from the skins of larger boney fish. It has a higher tack on first contact than hide glue. With Victorian chemistry it was also possible to make a cold fish glue that stayed liquid, but this wasn't possible for hide until fairly recently.