The axe and the cellar offer some vivid imagery. We can imagine that in a story like "The Black Cat," going to the cellar is a bad idea. We can almost smell the musty cellar smell wafting up at us. We might even experience a slight clouding of vision as the narrator, his wife, and the cat descend into the darkest depths of the "old" building (23). The imagery is vague and murky, until we get to the axe.
Like the knife, the axe has the potential to be used for violence. Most uses of the axe are violent, like chopping wood, for example. Firefighters use axes to save people, but the axe is still used violently to break things down. Here the axe is a symbol of the man's breakdown, and of the violent breakdown of his family.
If you see someone holding an axe, you might be slightly uncomfortable. You probably don't want one hanging about in your living room, either. When the narrator says he picked up "an axe," we think, "uh oh" (23).
We know what's probably going to happen next, especially if we've read Fyodor Dostoevsky's Crime and Punishment. When he says he "buried the axe in [his wife's] brain" our predictions come true (23). We are certainly horrified at the brutal murder of the woman. But, we might also be somewhat relieved that the cat managed to escape unscathed. As we discuss in "Writing Style" the narrator's fancy prose can hide meaning if we don't read carefully. Here, he's surprising blunt. Nothing fancy. Yet, this is one of the story's strongest images and we can understand it instantly.
Since the narrator keeps us in the cellar for the most of the rest of the story, we get walled up, or trapped, in the story. This speaks to our theme of "Freedom and Confinement." It also speaks to the narrator's trapped state of mind. Although he is free for a time to hurt others, the story shows him increasingly imprisoned. Everything comes together in the cellar – which is just one step away from the jail cell.
The importance of the cellar as a symbol, and as imagery, is ever more apparent when we look at it against some other aspects of the setting. If you're interested in this aspect of the story, be sure to take a look at "Setting."