The first thing we learn is that the nameless narrator is going to die the next day, and that he wants to write his story, which will be ugly. This story, the narrator says, is going to be about some things that happened to him at home. The "consequences" of what happened "have terrified – have tortured – have destroyed" him (1). We don't yet know why he's going to die the following day, or where exactly he is.
The narrator tells us that as a kid the he was a kind, sensitive animal lover. We also learn that he and his wife had had "birds, gold-fish, a fine dog, rabbits, a small monkey, and a cat" (3). The cat, of course, is Pluto. The conflict begins to unfold when the man describes the way his personality changed for the worse when he started drinking heavily, several years after Pluto became his pet. The conflict is within the narrator's home, between himself and his wife and pets, who he begins to abuse, physically and verbally, except for Pluto.
When the narrator turns on Pluto, he doesn't do it halfway. First he cuts the cat's eye out, and then he hangs him from the tree in his garden – leaving the body there when he goes to sleep. This definitely complicates things for the narrator. He is now a cat murderer, and his once happy home seems to be more and more nightmarish, especially for the other characters.
Somehow, when the narrator goes to sleep that night (after murdering Pluto in the morning) his house catches on fire. Someone (it's never revealed who) wakes him from his sleep with a warning, just in time. The narrator, his wife, and "a servant" escape the flames. All the family's financial security goes up in smoke. Presumably, the birds, gold-fish, […] fine dog, rabbits, [and] small monkey perish in the flames, though the narrator never mentions them again (3). The climax propels this desperate family into poverty and into changing residences.
As we discuss in "What's Up with the Title?" we can think of the second cat as either a modified version of Pluto, or a completely different cat. In any case, the arrival of the second cat marks the halfway point in this story. It is suspenseful precisely because we aren't sure what the second cat is. If the narrator can be believed, the cat is not only missing an eye, like Pluto, but also grows an image of a gallows on his chest (a "gallows" is an apparatus used for hanging people). The cat also seriously gets on the narrator's nerves. We might see the cat as affectionate, and desperate for affection, but the narrator sees him as executing some awful plot against him. In the stage we see the narrator getting worse and worse. And we learn that the narrator is writing from a "felon's cell" (20). Waiting to see what lands him in jail adds another layer of suspense to the story.
During that fateful trip to the cellar of the family's new residence (an "old building") the narrator tries to kill the cat with his axe. When his wife intervenes, the axe is turned on her. The narrator thinks he's successfully hidden the body and bluffed the cops. He isn't upset about killing his wife, and is happy he has managed to make the cat run away.
In the conclusion, the cat reappears, and the murder is discovered. The man seems convinced that the cat exposed him on purpose. The description of the cat's "voice" coming from inside the wall suggests that if the cat did intentionally allow himself to be walled up, in order to expose the man, he paid an awful price for it. Check out "What's Up with the Ending?" for a deeper analysis of this moment, and some other aspects of the ending.