Pet play is a variety of BDSM where one or more partners (usually the submissive) take on the role and/or features of a pet or other animal. Common incarnations include puppy play, kitten play, and pony play.
A furry is someone who identifies as a particular animal. Some of them dress up in tails, ears, paws, and/or full fur suits. It can have a sexual element, as much as any aspect of identity does, but is often just a matter of a deep communion with a particular species.
To the uninitiated, pet play and furdom might be indistinguishable: the trappings of both include various simulated animal parts and weird urges toward bestiality. (Actually, both furdom & pet play have nothing to do with bestiality. People who want to have sex with animals generally just have sex with animals.) To me, being a pet is very different from being a furry because instead of feeling like I myself am a puppy (a matter of identity) it is an expression of how I relate to my Master in scenes (a matter of role).
J started calling me Puppy after I repeatedly jumped on him in delight when he got home from work. The name stuck, and soon afterwards I had a collar and everything. As far as BDSM-style roles go, it’s a good one to have because it’s not nearly as serious as being a Gorean slave, but the feeling of subjugation & humiliation gives Gor a run for its money. It also allows room for romance (in my twisted definition) and cuddliness. I love everything about being a pet: the petting, the cuddling, the love, the playfulness, and when necessary, the punishment.