Duffy was an Earth dog that was born in 2132 and died thirteen years later, in 2145.
In 2152, Captain Jonathan Archer had a dream in which Porthos had died, and was being buried in a pet cemetery. Duffy and Goldie were among the gravestones also in the cemetery. (ENT: "A Night in Sickbay")
Although not explicitly stated, it is probable that Duffy was Jonathan Archer's dog, as he said that he couldn't remember not having a dog, and that he'd gotten Porthos only a couple years prior.