One theme of "The Black Cat" is that you can never escape guilt, it follows you for the rest of your life and often results in consequences. In this case, the guilt following you is the black and white cat. It also resembled evil considering that the color black is commonly known as an evil color. The purpose of the black and white cat following him was to mock him and remind him of what he had done to Pluto. Considering the fact that towards the end of the story, rhe black and white cat never left shows just how guilt can do the same -- follow you and never leave.
Another theme of "The Black Cat" is that the evil inside of you can result in evil sins. In this case, the "evil inside" is portrayed as alcohol. The character's alcoholism leads him to do evil sins such as hang his cat as well as murder his own wife.
Another theme of "The Black Cat" is that hiding your sins is only temporary. In this case, the narrator tried to hide his murder in the walls of the cellar. This, however, is only temporary because towards the end of the story, the black and white cat (a.k.a his guilt) reappears to mock him.