Horror stories usually have some pretty common elements: Violence, the supernatural, dead things never quite dying, trustworthy characters turning out to be untrustworthy, and murder. "The Black Cat" has all of those elements, to be sure. We have a man who, at the beginning, is a kind, loving husband and caregiver to animals. By the end he is abusive and violent, even killing his wife unexpectedly, and wreaking havoc on his pets. He is violent, and the story has some very gory details to it in describing the black cat's demise, and also his gruesome murder of his wife. There is the supernatural through the "visions" the man has of the cat, and also in the end as the cat is apparently not dead, but alive and well in the wall. All of these factors lend themselves well to the horror genre.
One last element of the horror genre that the story has is a main character with a very interesting psychology. At its heart, the story is really a tale of a man who, through the abuse of alcohol, becomes victim to the demons of his mind and his vices. It is written through the narrator's eyes; he himself describes his slow descent into madness and violence. He can explain, quite lucidly at a later date, what happened to him. His mind starts playing tricks on him; he doubts his sanity, and it leads him to do some pretty awful things. When we are unsure of our own mental soundness, everything is strange and bizarre, and we get to see that through the eyes of the narrator himself. His psychology is quite interesting, and despite his descent into moral decay, he relates the events of the story quite clearly after the fact.
I hope that helps; good luck!