The narrator of Edgar Allan Poe's "The Black Cat" explicitly states that the second cat reminds him of Pluto, the first cat that he mistreated and killed. However, as the narrator spends more time with the second cat, it also begins to represent certain incorporeal aspects of the narrator's own character.
Firstly, the two cats share a number of similarities. Like Pluto, the second cat shows a fondness for the narrator. To his surprise, the narrator soon finds "a dislike to it arising within [him]" due to this. He continues,
By slow degrees, these feelings of disgust and annoyance rose into the bitterness of hatred. I avoided the creature; a certain sense of shame, and the remembrance of my former deed of cruelty, preventing me from physically abusing it. I did not, for some weeks, strike, or otherwise violently ill use it; but gradually -- very gradually -- I came to look upon it with unutterable loathing, and to flee silently from its odious presence, as from the breath of a pestilence.
Because the cat reminds the narrator of Pluto, it reminds him of what he did to the first cat as well. This connection is furthered by how nearly identical the new cat is to Pluto, as the very next day after the narrator brings the cat home he discovered "that, like Pluto, it also had been deprived of one of its eyes." With the exception of a patch of white on its chest, it is nearly identical to Pluto. The wound elicits sympathy in the narrator's wife, which eventually contributes to the wife's murder later in the story. The cat continues to show strong affection toward the narrator, and he grows more and more angry at this, calling its caresses "loathsome." He feels the desire to kill it, just as he killed Pluto, but is stopped by his "absolute dread of the beast." Unlike with Pluto, and perhaps even because of what he did to Pluto, the narrator feels afraid of the creature.
There are a number of reasons why the narrator is reminded of Pluto. However, as the story progresses, the narrator begins to focus in on the one discernible difference between the cats, stating:
My wife had called my attention, more than once, to the character of the mark of white hair, of which I have spoken, and which constituted the sole visible difference between the strange beast and the one I had destroyed.
Over time, this patch begins to take on a more specific form in the narrator's mind:
The reader will remember that this mark, although large, had been originally very indefinite; but, by slow degrees -- degrees nearly imperceptible, and which for a long time my Reason struggled to reject as fanciful -- it had, at length, assumed a rigorous distinctness of outline. It was now the representation of an object that I shudder to name -- and for this, above all, I loathed, and dreaded, and would have rid myself of the monster had I dared -- it was now, I say, the image of a hideous -- of a ghastly thing -- of the GALLOWS ! -- oh, mournful and terrible engine of Horror and of Crime -- of Agony and of Death!
Through the narrator's meticulous focus on the mark, it eventually takes on the shape of the gallows in his mind. Like many of Poe's narrators, the man is most likely projecting. In this case, the narrator is projecting his feelings of guilt for his previous crime, and perhaps even for the crime he is about to commit, onto the cat, which looks very much like the victim of an earlier crime.
The second cat represents a number of things to the narrator: an earlier mistreatment, an earlier victim, and perhaps, through the patch of white, even the narrator's future act for which he is arrested and (probably) sentenced to death. All of these acts represent the narrator's guilt, even if, in true Poe fashion, he is unable to recognize and accept it for himself.