What symbols are used in Poe's "The Black Cat" and how does he establish fear and horror?

Poe establishes an atmosphere of fear and horror in several different ways in "The Black Cat." Firstly, the narrator, who is telling the story from the first person point of view, describes himself as docile, humane and a lover of pets. But this character ends up doing the most unimaginable things. He abuses, harms and kills his favorite pet and later ends up killing his wife. Perhaps what is most horrific about the story is the ease with which the narrator describes his deeds. 

The narrator is frequently contradicting himself. He used to be docile but later turned murderous. He had regret and even shame about harming his cat but later feels fine about having murdered his wife. The fact that he can commit these dreadful deeds in cold-blood is very disturbing.

As for symbols, most readers agree that there are many symbols associated with the two cats in the story. The first cat is a completely black cat. In European history, black cats were thought to be unlikely, some even thought that they were witches. In fact, the narrator mentions that his wife said the same about the cat jokingly. This is an allusion to the historical myth about black cats. The narrator tries to give the impression that he is not superstitious and doesn't believe these stories. But as he develops a progressively hateful attitude toward this cat that he once loved, one wonders whether he truly believed the myth. (By the way, black cats are in fact lucky as the gene that makes them all black [called melanism] also reduces their risk of disease. Cats with melanin are the opposite of albino cats that are all white. The reason for their color is the same, it has to do with the amount of melanin in their skin and hair).

The other symbolism about the first cat is his name, Pluto. In Greek mythology, Pluto was the ruler of the dead and the underworld. So perhaps his name was a premonition of his fate? Or perhaps his name convinced the narrator of his doubts about the cat and whether the cat had something to do with witchcraft.

There is also symbolism associated with the narrator's second cat, which looked a lot like Pluto. He was black, except with a patch of white on his chest. Like Pluto, he too was missing one eye. Although the narrator doesn't express it very directly, he seems to dislike this cat in time because of his resemblance to Pluto. Maybe the narrator hated this cat like he hated Pluto. Maybe he hated this cat because it reminded him of the terrible thing he had done to Pluto (carved his eye out and later killed him by hanging him off a tree). The fact that the narrator believes that the white patch on the cat is changing in appearance to look like the gallows supports the narrator's growing guilt over his sin. 

The narrator foreshadows various things in the story, such as that he will be violent towards his wife. Seeing gallows in the cat's fur can be both about guilt and also a foreshadow of the future that awaits him.

Some also associate the cats with the narrator's wife. Like his wife, they are loving, nice animals. Despite the narrator's alcohol abuse and ill treatment of his wife, his wife is patient and never complains. And his wife is also fond of animals and especially loves the cats.

We must also mention that the narrator appears to be highly disillusioned. He develops a sudden and difficult to understand dislike towards his pets. He had taken both cats willingly and had liked them in the beginning. But he finds them fear inducing and horrific after some time. The narrator seems to imply that it's because of his alcohol abuse. He increasingly finds some kind of meaning in random things. He seems to be struggling with his understanding and perception of what is happening. He tries to find logical explanations for incidents and even finds ways to excuse his behavior. But it's obvious that he is walking on a thin line of madness.

What symbols are present in Edgar Allan Poe's "The Black Cat"?

A symbol is...

...an object [that] is meant to be representative of something or an idea greater than the object itself.

In Edgar Allan Poe's story, "The Black Cat," the first symbol is found in the black cat. They are symbolic of evil—often believed to be a witch's familiar—a creature that served the witch and her evil purpose (a belief the narrator's wife also had).

In this tale, the first black cat—Pluto—is symbolic of the deterioration of the narrator's mind. 

Pluto is symbolic of change and transformation.

The narrator speaks candidly about how he changed:

I grew, day by day, more moody, more irritable, more regardless of the feelings of others. I suffered myself to use intemperate language to my wife. At length, I even offered her personal violence. 

Soon, his anger and violence are visited upon his pets, including Pluto.

I seized [the cat]; when, in his fright at my violence, he inflicted a slight wound upon my hand with his teeth. The fury of a demon instantly possessed me. I knew myself no longer...and deliberately cut one of its eyes from the socket!

(In classic mythology, Pluto is the name of the god of the underworld: Hades. As time goes by, Pluto certainly seems to reside in his own form of hell.)

The cat heals, with one eye that functions. This eye brings to mind Poe's "The Tell-Tale Heart." In terms of that story ("watchfulness"), and the narrator's shame over what he has done, I believe the cat's eye is symbolic of sin: specifically, the sin of the narrator, and the evil that drove him to harm Pluto—finally hanging the cat from a tree:

...[I] hung [Pluto] because I knew that in so doing I was committing a sin—a deadly sin that would so jeopardize my immortal soul as to place it—if such a thing were possible—even beyond the reach of the infinite mercy of the Most Merciful and Most Terrible God.

The second black cat that "attaches" itself to the narrator is symbolic of guilt. It is, in many ways, very much like Pluto once was: black, affectionate and lacking one eye. It pursues the narrator without ceasing:

With my aversion to this cat, however, its partiality for myself seemed to increase. It followed my footsteps with a pertinacity which it would be difficult to make the reader comprehend. Whenever I sat, it would crouch beneath my chair, or spring upon my knees, covering me with its loathsome caresses. 

Once the narrator murders his wife, the second black cat seems to symbolize justice. For it is the cat, being walled up with the corpse of the narrator's dead wife, that alerts to the authorities to the place the man has hidden his wife's body. (He refers to the cat as the "Arch-Fiend," as if it purposely betrayed him.)

No sooner had the reverberation of my blows sunk into silence than I was answered by a voice from within the tomb!—by a cry, at first muffled and broken, like the sobbing of a child, and then quickly swelling into one long, loud, and continuous scream, utterly anomalous and inhuman—a howl—a wailing shriek, half of horror and half of triumph, such as might have arisen only out of hell, conjointly from the throats of the damned in their agony and of the demons that exult in the damnation.

The narrator's personification of the cat...

...a wailing shriek, half of horror and half of triumph...

...provides the reader with a sense that the cat has human intelligence, giving it more power to affect the story's outcome in light of the personification.