The antagonist of a story is the character who is in conflict with the protagonist, and the protagonist is the main character of the story. The main character in "The Black Cat" is the narrator, and although the narrator is clearly in conflict with himself throughout the story, he is also, from his perspective, in conflict with his pet cat.
At the very beginning of the story, we are told that the narrator and his black cat initially get on very well together. The cat becomes the narrator's "favorite pet and playmate." However, one night, coming home drunk, the narrator decides to cut out one of the cat's eyes. From that moment on, understandably, the cat becomes wary of its owner. Not long after, the narrator decides to kill his cat.
One interpretation of the events following this grisly act is that, after the cat's death, the spirit of the cat begins to relentlessly haunt the speaker. It is thus the cat's death which marks the moment from which the narrator and the cat are in real conflict. To the narrator, the spirit of the dead cat becomes the antagonist and is no longer merely the victim.
At first, the dead cat appears to the narrator only as a disconcerting "apparition" on a wall, but then, in order to better torment him, it possesses the body of a second, almost identical cat. In the form of this second cat, the spirit of the first cat haunts and torments its former owner, inspiring in him unbearable feelings of "terror and horror." The narrator comes to know the second cat as a "brute beast" and his "tormentor."
Eventually, the cat gets its revenge on the narrator and thereby fulfills its role as the antagonist of the story. It is the cries of the cat which alert the police officers to the whereabouts of the narrator's murdered wife and which thus condemn the narrator to his death.