What are three different types of Gothic elements of Literature that are used in "The Black Cat," and quotations to support them?
I would start off by observing that this excellent story uses a classic hallmark of Gothic literature: the unreliable narrator. This is of course a Gothic element that Poe uses elsewhere in his shorter fiction to great effect in stories such as "The Tell-Tale Heart." Consider how the narrator introduces himself and tries to present himself as a reasonable, thining individual:
For the most wild, yet most homely narrative which I am about to pen, I neither expect nor solicit belief. Mad indeed would I be to expect it, in a case where my very senses reject their own evidence. Yet, mad I am not--and very surely do I not dream.
Poe deliberately makes us question the reliability of such a narrator and whether indeed the credibility of the narrator given what he tells us about himself and his drunkenness can be trusted.
Secondly, there is reference to maddened rage and violence, especially in the acts of anger perpetrated by the narrator. Consider how he reacts to being slightly bitten by the cat:
The fury of a demon instantly possessed me. I knew myself no longer... I took from my waistcoat-pocket a pen-knife, opened it, grasped the poor beast by the throat, and deliberately cut one of its eyes from the socket!
The excess of emotion and anger that is here refered to is a key feature of Gothic literature, as we enter the dark realms of the psyche of man and see all the horrendous deeds and acts he is capable of doing.
Lastly, consider the supernatural elements of this story, in particular in the way that Pluto is somehow "resurrected" in the form of a new cat which, like Pluto, lacks one of its eyes. The only difference is the white mark which represents the "gallows." This, combined with the figure of the "gigantic cat" that is left on the wall after the fire and the way in which the cat is shut up with the body of his dead wife clearly hints at supernatural happenings that are beyond our reason or comprehension.