Understanding how irony in "The Black Cat" works requires a basic understanding of the forms irony can take in literature. There are two standard ways of thinking about irony. The most easily identifiable form of irony occurs when the character or plot provides a storyline that is opposite to what the author ostensibly wants to convey. In modern literature, irony depicts how the elements of a story work together to show detachment from an emotion or experience.
The irony in "The Black Cat" occurs after the narrator cuts out one the eyes of his cat Pluto. Although the cat no longer can see with that eye, the cat now sees its caretaker for what he really is -- unpredictable and dangerous. Poe uses irony to indicate the cat's newfound insight and detachment from the narrator after seeing his true nature. It is ironic that when the cat had the use of both eyes, it didn't see the narrator's true persona; however, once the cat loses an eye, for the first time it sees its owner for who he really is.
The narrator in "The Black Cat" does not have a name. He tells his story for the purpose of recounting the events in a simple cause-effect relationship. The story begins as he talked about his youth and how his parents purchased all manner of pets for him. He was chastised as a child for having too tender a heart. Ironically, as he grows into an adult, we find that he becomes capable of psychotic deeds and incredible cruelty to his own pet Pluto and to those that are close to him.
A final, horrifying touch of irony happens when the narrator tries to murder his cat with an ax. She grabs the ax to stop him, assuming that she will be safe from harm. The narrator draws back the ax and splits his wife's head open with the ax, killing her instantly. The irony here is that while trying to save the life of the cat, she loses her own.