The black cat symbolizes the state of the narrator's soul-which is black, mutilated, and decaying. The black cat is symbolic because it is the cat's meowing that draws attention to the wall, and the perverse pleasure the black soul of the narrator takes in believing he has gotten away from it.
There are many ironies in this story. One is the way the narrator worries that after he relates his story, others will not give it much thought-it is just an ordinary event. Yet, the narrator is telling the story from his prison cell awaiting his death-and his tale is of his murderous rage. This seems very far from ordinary.
Other ironies include his description of the torture and mutilation of the cat as a "silly action" done for no real reason, and also the clam retelling of the murders of both his wife and the cat, and then going calmly on with his life.
The major irony is that his pompous arrogance is really what led to his being apprehended. He was so sure they would never discover his secret, and his nonchalant tapping of the wall leads the cat to wailing, and ultimately to his being caught.