There is no authoritative list of the roles in puppy play, but the most common names and meanings are these:
are more interested in the dog persona: barking, being on all fours, the physical aspects of being a dog.
are more about the canine head space then the physical, though physical aspects may and do come into it as well.
are the more aggressive, dominating pups. They will pose themselves as pack leaders, but will often also look after the pups in their pack.
are anyone the takes care of a pup. The one holding the leash per se.
train the pups to behave, do tricks. Trainer and Handlers may overlap.
are usually the same as in other BDSM roles, they own the pups, and may or may not take part in the other roles.