What does the second cat symbolize in The Black Cat?

I think it is possible to read the second cat as symbolic of the narrator's guilt. After he maims and murders Pluto, the first black cat, he says that "there came back into my spirit a half-sentiment that seemed, but was not, remorse." He says that it is not quite remorse, but he does begin to "look about [himself]" to find another cat, the same in appearance, to replace the first. The narrator likes the new cat at first, but he gradually comes to "dislike" it, and then it soon "disgusted and annoyed" him, and then he eventually grows to feel the "bitterness of hatred" for the cat. He fails to abuse it out of "shame, and the remembrance of [his] former deed" (murdering Pluto), and this sounds a great deal like guilt. The narrator's sense of guilt is vague at first—hard for him even to name—but it grows and begins to determine his behavior toward the cat. He admits that he "longed to destroy it" but holds back "partly by a memory of [his] former crime, but chiefly . . . by absolute dread of the beast." There is no reason to "dread" this cat if not for the feelings with which the narrator associates it; the cat seems to love him and is very affectionate. Perhaps the narrator dreads the cat because of what the cat represents, his own guilty conscience.

As his guilt seems to come into sharper focus, so does the noose of white fur on the new cat's breast. The narrator seems to feel guilty about hanging Pluto, and so the new cat now begins to manifest that guilt physically. The narrator feels that this "brute beast" is causing him "insufferable woe," but—again, the cat itself does nothing to cause this feeling, but the narrator's own guilty conscience, which he seems unable to recognize, could cause it. He says that he feels this woe to be "incumbent eternally upon [his] heart," strengthening the argument that it is guilt he feels, as this would weigh heavily upon him.

After the narrator murders his wife, a crime he calls "hideous" (an acknowledgement that seems to imply guilt), he hides her corpse in the wall, and he is relieved to see that the cat seems to have disappeared as well. He feels like he has gotten away with his crime, and he says that he "slept [tranquilly] even with the burden of murder upon [his] soul." With the cat gone, he seems to believe that he has escaped his guilty conscience, even saying that his "happiness was supreme!" and that "The guilt of [his] dark deed disturbed [him] but little"—though it does disturb him.  

Finally, the police come and search his home, and—finding nothing suspicious—they begin to take their leave. However, the narrator, inexplicably, feels a "rabid desire to say something easily" though he "scarcely knew what [he] uttered at all." The investigators were on their way out, and the narrator would have been home free if he would have kept his mouth shut, but he could not. He blathers on about the construction of the house, even striking the wall with his cane, and just as he could not keep his own mouth shut, the cat begins to "scream" and "howl" from behind the wall, giving the murderer away. The narrator's guilt compelled him to speak up just as he was about to get away with his crime, prompting the cat to "speak" too, symbolically linking the narrator's guilt with the second cat.

What do cats symbolize in "The Black Cat" and "The Cats of Ulthar"?

When drawing a comparison between H. P. Lovecraft's "The Cats of Ulthar" and Edgar Allen Poe's "The Black Cat," I would specifically refer you to the first paragraph of Lovecraft's short story, which establishes in cats a supernatural quality. To give one particular sentence that stands out here:

For the cat is cryptic, and close to strange things which men cannot see. (Lovecraft, "The Cats of Ulthar")

This one sentence, I think, encapsulates much of the shared thematic ground of these two stories, for all that they have radically different settings and tones.

Poe's story is far more grounded and set in the real world, and there is a lingering question as to the degree to which the entire series of events exists primarily within the narrator's own head, or whether there is, in fact, some supernatural explanation in effect. Even so, this ambiguity is noteworthy because the rest of the story is so grounded in reality. But with the cats, questions emerge as to how much of this is a creation of the narrator's own inner turmoil and how much of it is the work of something supernatural.

Lovecraft's "Cats of Ulthar" is far less grounded in its tone and setting (this story is purely a work of fantasy), but here too, we see cats connected with the supernatural. Like in "The Black Cat," the story centers around acts of violence perpetrated against cats, along with punishment for that violence. Additionally, much as in "The Black Cat," Lovecraft leaves many of the details of his story purposefully ambiguous. For example, while we could say that Menes seems to have cast some sort of magic, we can't say much as to how that magic functions or how exactly it is connected to the behavior of the cats (assuming that it is even connected to the cats at all). These questions are all left unanswered. In both cases, the true nature of these cats is a mystery of the worlds they inhabit.

Cats symbolize wisdom and justice in both tales.  In Poe's "Black Cat," the narrator begins by describing his love for creatures.  But, "the disease of alcohol" begins to consume him, and he takes it out on his cat which seems to return later in the form of an identical cat who will not let the narrator forget his cruel acts toward his pet.  In the end, the narrator thinks that he has gotten away with murdering his wife and hiding her body within the wall of his home, but the police hear the cat in from behind the wall and discover the narrator's dead wife.

In Lovecraft's "The Cats of Ulthar," the narrator stresses that

"in Ulthar, which lies beyond the river Skai, no man may kill a cat."

Thus, when a cotter and his wife kill cats, especially the prized cat of a little orphan boy, the cats get revenge upon the couple by attacking and eating them at night.

In both stories, the cats are black and seem to be immortal (similar to Egyptian beliefs).  They try in their own way to provide warnings to their master or oppressor, but neither listens, and justice is served when the cats get revenge for the wrong done to them or to others--hence, the fear of cats.