Monster is a dog that the Foutley family owned before he ran away.
He may have ran away from home just before the series began. As a result, Carl and Hoodsey used his doghouse for planning their schemes and also kept all of their grotesque items. When he is seen laying on the porch of his new owners' house in The Wedding Frame, he picks up Carl's scent and goes looking for him. He and Carl are happily reunited at Lois and Dr. Dave's wedding.