Horror or Gothic fiction can also be "sensation" fiction, a popular genre in Poe's day. Sensation stories were designed to work on the readers' senses. The characters experience scandalous feelings and desires, commit dire acts, and find themselves in extreme situations. Violence, imprisonment, death, dismemberment, and live burial are meant to shock us into feeling.
In "The Black Cat," most of the drama occurs in the home, and revolves around the narrator's relationship with his wife and pets. Like so many narratives of terror and depravity, this one combines the family drama with the horror or Gothic. This taps into some of our deepest fears – fears of what can go wrong at home. Home is where we are supposed to be most comfortable and safe, and is also where we are most vulnerable. As in " The Tell-Tale-Heart," however, home for the characters in this story is anything but safe.
You're probably starting to see why we call this a psychological thriller. It tries to get into our heads, but also plunges us inside the twisted mind of an abusive killer detailing the stages of his breakdown. We keep reading because we want to see what happens next, and because we want to find out why or how this guy got so twisted. The suspense of "The Black Cat" extends beyond the end of the story, since the ending doesn't give us many answers. The possibility that supernatural forces are at work adds another layer of suspense, and, perhaps, another layer of terror.
The Southern Gothic is a sub genre of the Gothic. It usually features a southern setting, and deals with issues of slavery, or the American South after slavery. Poe was writing before the Civil War. As you know, the institution of slavery was still legal in the US, and was a part of southern life. Critics and readers have been debating Poe's stance on slavery and race for a long time. Because Poe was virtually silent on the issue, this debate will most likely continue for a long time.
Several essays in the collection Romancing the Shadow: Poe and Race discuss the possibility that "The Black Cat" might be the story of hanged slave, in addition to being the story of a hanged cat. If you find this theory credible then you'll agree that this story is Southern Gothic. If not, then you won't. If you want to learn more about this theory, check out the "Character Analysis" of Pluto.