Thing varies according to continuity. In the original TV series continuity, Thing T. Thing was a hand with a forearm attached to it and apparently a member of a different species (as when he falls in love with a relative's "hand maiden", a female hand). He seemed capable of appearing out of any box or aperture, which led to popular fan speculation of the time that he was actually the extrusion into this reality of a multi-limbed Cthulhuan entity. (Few people recall that fan speculation ever since the film came out.)
In the film continuity, Thing is a hand without a forearm attached to it and apparently just another unique member of the Addams clan.