There's clearly some kind of metallic claw or gauntlet over his left hand. I think Hojo's experiments might've damaged or altered that limb, maybe weakened it to the point where he needs a glove to protect it.
He definitely had a human hand back before Hojo's experiments, so it's some kind of by-product of that procedure.[q=Dante Wolfwood]1.Hojo added an animal claw to him as he used him as a guinea pig for jenova injections.[/q]Most people agree that Hojo's experiments on Vincent didn't involve Jenova cells. If that had been the case, Vincent would likely have suffered delusions or loss of control, similar to Cloud and the other 'clones'. Hojo wanted to keep Vincent away from the Jenova project, in any case. Whatever was done to him, it involved some other kind of tranformation - something to do with monsters, perhaps. In the FFVII world, the number of monsters began to increase when Mako reactors became widespread; perhaps Vincent's transformations are something similar?[q]3.Everytime he transforms, He loses a part of himself everytime and his body is slowly completely transforming into a mindless monster.(Hey! Maybe a theme in FFVIIC! who knows)[/q]But then, he had that claw from the moment the experiments were finished, and never noticeably changed during the game...

Maybe AC and DoC will clear up this question, and others. But for now, I believe that his clawed glove is something to do with the experiments that made his body capable of demonic transformations, though I don't think a simple piece of clothing could be the cause of so much. He keeps most of his body hidden, only his right arm and part of his face are visible, most of the time. Incidentally, the profile in the game's instruction manual says that "he may seem frail at first, but inside his body lurks a fearsome power".