Trainability
Intelligence (but not the point of it being destructive if it's bored)
Ease of grooming (I prefer short-haired animals)
Calm temperament
A sweet facial expression
Relatively small size. I don't want any enormous animals around that are going to eat me out of house and home. I have enough trouble with that having my husband and son in residence.
A protruding snout such as is seen on a tabby cat or a Labrador retriver as opposed to a pushed-in face like is seen on a Persian cat or a pug or bulldog.