Here's an answer straight from Kermit's mouth on what he is. I believe you can extrapolate that out to the other Muppets as well:
Weekend Update with Kermit the Frog
Meyers: ... which means, in this case, Congress was nothing more than a puppet to the food industry. (to Kermit) No offense...
Kermit: Oh that's okay I'm not a puppet I'm a Muppet.
Meyers: Oh what's the difference?
Kermit: Ah well, a puppet is actually controlled by a person, whereas I am an actual talking frog.
