Exposition- The narrator is presumably in jail and awaiting to be executed. He sets the tone and mood of the novel with his anxious talk, where he says that something terrible has happened, and also that something has terrified him.
The conflict of the story is basically that the narrator has a massive drinking problem. The way that this is conveyed is when he explains that, after getting his pet- a black cat named Pluto- he had developed a tendency to drink heavily. As a result, he is abusing his pets and even his wife, but not Pluto--at least not yet
[…] through the instrumentality of the Fiend Intemperance [I] had (I blush to confess it) experienced a radical alteration for the worse.
The rising event, which is what will lead to the climax (also known as the complication) occurs when, during one of the many fits of temporary madness that the narrator endures after drinking, he picks on Pluto.
My original soul seemed, at once, to take its flight from my body; and a more than fiendish malevolence, gin-nurtured, thrilled every fibre of my frame
He first tortures him by taking off its eye. Then he hangs the cat. This horrible scene sets the scenario for something even more terrible, as now it is clear that the narrator has lost control of himself and, eventually, of his fate.
The climax then occurs shortly after this, because "something" or "someone" set the narrator's house on fire. This means that all of the belongings of the family are also burned and they all become destitute. This could be seen as the afterlife revenge of Pluto, but we do not know that yet.
The falling action, leading to the final denouement comes with the entrance of another black cat to the story. We do not know if this is a Pluto evil comeback. We only know that this further complicates the situation for the narrator, as now he is being literally haunted by a living being.
The denouement is the murder of the wife. The narrator, who has clearly lost his mind had not yet told why he was in jail in the first place. Now we know that, during one of his persecution moments he kills his wife (he intended to kill the cat) with an ax and so he hides the body. The narrator confesses that he does not care so long as the black cat does not return.
The conclusion is that he is found out and the crime is exposed because the sounds of a meowing cat were heard coming from the wall. It could be considered divine justice or perhaps part of the insanity of the narrator. That is up to the reader to figure out.