A symbol is...
...an object [that] is meant to be representative of something or an idea greater than the object itself.
In Edgar Allan Poe's story, "The Black Cat," the first symbol is found in the black cat. They are symbolic of evil—often believed to be a witch's familiar—a creature that served the witch and her evil purpose (a belief the narrator's wife also had).
In this tale, the first black cat—Pluto—is symbolic of the deterioration of the narrator's mind.
Pluto is symbolic of change and transformation.
The narrator speaks candidly about how he changed:
I grew, day by day, more moody, more irritable, more regardless of the feelings of others. I suffered myself to use intemperate language to my wife. At length, I even offered her personal violence.
Soon, his anger and violence are visited upon his pets, including Pluto.
I seized [the cat]; when, in his fright at my violence, he inflicted a slight wound upon my hand with his teeth. The fury of a demon instantly possessed me. I knew myself no longer...and deliberately cut one of its eyes from the socket!
(In classic mythology, Pluto is the name of the god of the underworld: Hades. As time goes by, Pluto certainly seems to reside in his own form of hell.)
The cat heals, with one eye that functions. This eye brings to mind Poe's "The Tell-Tale Heart." In terms of that story ("watchfulness"), and the narrator's shame over what he has done, I believe the cat's eye is symbolic of sin: specifically, the sin of the narrator, and the evil that drove him to harm Pluto—finally hanging the cat from a tree:
...[I] hung [Pluto] because I knew that in so doing I was committing a sin—a deadly sin that would so jeopardize my immortal soul as to place it—if such a thing were possible—even beyond the reach of the infinite mercy of the Most Merciful and Most Terrible God.
The second black cat that "attaches" itself to the narrator is symbolic of guilt. It is, in many ways, very much like Pluto once was: black, affectionate and lacking one eye. It pursues the narrator without ceasing:
With my aversion to this cat, however, its partiality for myself seemed to increase. It followed my footsteps with a pertinacity which it would be difficult to make the reader comprehend. Whenever I sat, it would crouch beneath my chair, or spring upon my knees, covering me with its loathsome caresses.
Once the narrator murders his wife, the second black cat seems to symbolize justice. For it is the cat, being walled up with the corpse of the narrator's dead wife, that alerts to the authorities to the place the man has hidden his wife's body. (He refers to the cat as the "Arch-Fiend," as if it purposely betrayed him.)
No sooner had the reverberation of my blows sunk into silence than I was answered by a voice from within the tomb!—by a cry, at first muffled and broken, like the sobbing of a child, and then quickly swelling into one long, loud, and continuous scream, utterly anomalous and inhuman—a howl—a wailing shriek, half of horror and half of triumph, such as might have arisen only out of hell, conjointly from the throats of the damned in their agony and of the demons that exult in the damnation.
The narrator's personification of the cat...
...a wailing shriek, half of horror and half of triumph...
...provides the reader with a sense that the cat has human intelligence, giving it more power to affect the story's outcome in light of the personification.