The narrator of "The Black Cat" kills his wife because she dares to prevent him from killing their pet cat. As he goes down into their cellar one day, she goes with him, and the cat nearly trips him on the steps. In the past, his fearfulness of the animal prevented him from harming it, but this sent him into a rage. Holding an axe, he lifts it and aims a blow at the cat in order to kill it, but his wife stays his hand. Already "exasperated ... to madness" by the fact that the cat nearly tripped him and sent him "headlong" to the bottom of the stairs, he then swings the axe down onto his wife's head, burying it "in her brain." She falls dead immediately.
All she did, then, was attempt to prevent the narrator from killing their cat with an axe—which seems a perfectly reasonable thing to try to keep someone from doing—and so he murdered his wife instead. After the fact, he does not even express remorse for what he did. He quickly figures out what to do with her body so that his guilt will not be detected, and he makes plans to wall her corpse up behind some bricks and plaster in their basement. He deceives the police officers who come to investigate his wife's disappearance, and he very nearly gets away with his crime, which is revealed by the black cat.