Symbolism, Imagery, Allegory
The dog, I think I told you, is a black monster of a beast: an oversized head, tiny, tiny ears, and eyes…bloodshot, infected, maybe; and a body you can see the ribs through the skin. (157)
That's the landlady's dog Jerry is talking about; a giant, evil beast with a red "erection" (157) and a foul temper. The dog attacks Jerry, Jerry tries to poison him. It's an epic struggle, smack dab in the middle of the play. It's the longest speech in the book. So what's the deal with the dog?
You're probably thinking the dog symbolizes some sort of existential whatsit about the horror of humanity and the cruelty of life.
And that's more or less right. Wow, good guess.
The title of The Zoo Story (see "What's Up With the Title?") indicates that humans are animals—"You're an animal too," Jerry tells Peter at the end of the play (278). So Jerry's struggle with the dog is in a lot of ways his struggle with other people. It's not just the dog that is mean, confusing, and bitter—it's other people. And Jerry's efforts to communicate with the dog, first by gentleness and then by violence, might as well be Jerry's efforts to communicate with Peter:
If you can't deal with people, you have to make a start somewhere. WITH ANIMALS! (163)
It's significant too that Jerry ends up coming to a kind of "understanding" (165) with the dog through violence. The dog tries to bite him, he tries to poison the dog, and then they both avoid each other like two people after a bad break-up:
It's very sad, but you'll have to admit that it is an understanding. We had made many attempts at contact and we had failed. (165)
And this seems like what Jerry tries to reach with Peter, too. He talks to him, tries to reach out, and when Peter doesn't understand, Jerry resorts to violence as a form of connection. At the end he seems to think he's succeeded—"you have comforted me. Dear Peter" (274). Comfort that leaves you bleeding to death on a bench, though, seems like comfort you could do without.