There can be a few reasons why a bird might lose its tail. It could be a natural part of that bird's molting cycle, or the tail feathers could have been shed because of illness or parasites. It could even have been pulled off by a predator the bird was fortunate enough to escape from or be a genetic anomaly. For this bird specifically, there are no strong indicators as to why he doesn't have his tail feathers. While he does show a touch of swelling around the eye (I regularly have outbreaks of house finch eye disease - avian conjunctivitis - at my feeders, despite my best efforts to keep the feeders clean), losing tail feathers isn't a typical symptom of that disease. He has no other signs of illness, and the condition of the rest of his plumage indicates he isn't molting. If he had a close call with a predator, he shows no other signs of it. All of those factors considered, and given that he's well able to fly and isn't lacking in any way, I'm inclined to believe this may be a genetic condition for this bird.
Still, it is interesting to see, and more interesting still when those unusual conditions help us learn to recognize individual birds. Terry the tail-less house finch has been around for a few days off and on, feeding with the other members of my house finch flock without detriment. Tail or no tail, he's as welcome as any.