What is the theme of "The Cask of Amontillado," and how does the story embody it?

The theme of Poe's wonderful story is revenge, though I'd probably narrow that a bit: the overwhelming intensity of revenge and how it can distort and destroy lives. The story relates to this theme on both the literal and the psychological level. Literally, Montressor bricks Fortunato into the walls of the catacomb and kills him. On the psychological level, the mysterious "thousand injuries" that Fortunato has done to Montressor have unbalanced him, turning him from a harmless wine snob to a killer.
Greg

What is the theme of "The Cask of Amontillado," and how does the story embody it?

The theme of "The Cask of Amontillado" is the execution of a perfect crime. The whole story is taken up with Montresor's description of how he did it, not why he did it. It might be called a "howdunit" rather than a "whodunit." Poe makes this clear at the outset because he glosses over Montresor's motive with a single sentence:

THE THOUSAND INJURIES of Fortunato I had borne as I best could, but when he ventured upon insult I vowed revenge.

That is all in the past. The story is concerned only with the present. This makes it easy to visualize, almost as if it were a motion picture. The other important sentence in the opening is:

At length I would be avenged; this was a point definitely settled—but the very definitiveness with which it was resolved precluded the idea of risk.

The motive and the decision to be avenged are both "settled." What remains is to carry out the murder. This is a very adroit way of handling the story. Poe only needs to focus on one problem, which is the logistical one of luring Fortunato to his palazzo and down into his catacombs. The reader is not distracted by other considerations. The story has immediacy. Something could go wrong. Fortunato could start asking questions or balking at being led so far. Montresor has enough problems in the immediate present without needing to deal with whatever must have happened in the past.

The story ends almost as soon as Montresor finishes building his stone wall. Fifty years have passed since he completed it, but he does not say anything about what he has been doing for an entire half-century. Poe whisks the story to a conclusion with just a few words:

Against the new masonry I re-erected the old rampart of bones. For the half of a century no mortal has disturbed them. In pace requiescat!

So the past is disposed of quickly and the future—what happened after he finished building the wall—is disposed of equally quickly. Poe focuses on the little time period between the men's encounter on the street and the completion of the "immolation." Virtually everything takes place in the "present," and virtually everything can be told in description and dialogue, as in a play or movie.

The theme, of course, is revenge, but the story is concerned only with the execution of that revenge. Poe hardly even touches on Montresor's feelings about Fortunato and says virtually nothing about their relationship. This is strongly suggestive of what Ernest Hemingway was to say about his "iceberg principle" many years later.

If a writer of prose knows enough of what he is writing about he may omit things that he knows and the reader, if the writer is writing truly enough, will have a feeling of those things as strongly as though the writer had stated them. The dignity of movement of an ice-berg is due to only one-eighth of it being above water. A writer who omits things because he does not know them only makes hollow places in his writing.

—Ernest Hemingway, Death in the Afternoon

What is the theme of "The Cask of Amontillado," and how does the story embody it?

In “The Cask of Amontillado," Edgar Allen Poe’s powerful tale of revenge, the reader sees how an obsession with vengeance can overwhelm a person.

When we meet the narrator, Montresor, he is vowing revenge upon Fortunato.

The thousand injuries of Fortunato I had borne as I best could, but when he ventured upon insult, I vowed revenge.

He tells readers he’s been wronged, but he never reveals the nature of the injury. All we know is that Montresor has been preoccupied with these wrongs. But what Fortunato has actually done is immaterial; Poe’s point is how revenge can drive a person to extremes. Montresor declares:

I must not only punish, but punish with impunity. A wrong is unredressed when retribution overtakes its redresser. It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong.

With this statement, we see how powerful Montresor’s need for revenge is. He seems to understand that revenge is a force powerful enough to “overtake” the party who seeks it, but he is too fixated on his need to recognize that his mind has become a slave to this passion.

Because he never actually reveals the actual insult, Poe illustrates that the narrator’s vindictiveness is not really connected to Fortunato’s crime. It is more closely linked to Montresor’s family legacy of vengeance. His family motto is "Nemo me impune lacessit,” or "No one assails me with impunity." The coat of arms bears a

... huge human foot d'or, in a field of azure; the foot crushes a serpent rampant whose fangs are embedded in the heel.

We can see from this image that Montresor has been reared by a family who put revenge at the center of their values, judging by how they chose to represent themselves on their coat of arms. Therefore, it makes sense that his plan to kill Fortunato is premeditated and well thought-out. He has located the right spot, made sure he has the tools of bricks and mortar available, and even figured out a tortuous and gruesome method to kill him. He will make him suffer the anguish of a slow death by watching Montresor wall him up alive, ignoring Fortunato’s screams. Then he will experience the agony of burning alive. This is a cold-blooded revenge, and fifty years later, the narrator feels no remorse. He ends the story with an icy, ironic, “In pace requiescat!” (Rest in peace!)

Another interesting point is that an integral part of the success of this plan is the way Montresor lures Fortunato to his doom. He appeals to Fortunato’s ego, telling him that he is on his way to find Luchesi, one of Fortunato’s rivals, who can determine if the wine Montresor recently bought is actually Amontillado.

"As you are engaged, I am on my way to Luchesi. If any one has a critical turn, it is he. He will tell me—"

"Luchesi cannot tell Amontillado from Sherry."

"And yet some fools will have it that his taste is a match for your own."

"Come, let us go."

The mention of his rival triggers Fortunato’s massive ego. Montresor, feigning concern over Fortunato’s cough and obvious illness, actually gives him several chances to turn back. But Fortunato insists on going forward because he cannot stand the thought that Luchresi could be on a par with him. His pride leads to his demise, which could be considered a secondary theme of the story.

What is the theme of "The Cask of Amontillado," and how does the story embody it?

You asked more than one question so I have edited your question to focus on this excellent gothic short story. Clearly a central theme that goes to the very heart of this excellent short story is that of revenge. From the very first paragraph it is clear that revenge is Montresor's central motive for acting in the way that he does:

The thousand injuries of Fortunato I had borne as best I could; but when he ventured upon insult, I vowed revenge.

Although the exact nature of this "insult" is never divulged, revenge is the driving force that leads Montresor to plot and enact his most gruesome of punishments. Consider Montresor's motto, "Nemo me impune lacessit", which means "Nobody attacks me without punishment", which adds a moment of black humour to the story as Fortunato responds to the firmness of this motto with a strong approbation: '"Good!" he said.' Of course, the first paragraph is key to the reader in terms of displaying Montresor's plan, which is an enactment of his motto:

A wrong is unredressed when retribution overtakes its redresser. It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong.

It is vital therefore to Montresor that Fortunato meets his fate at his hand - Fortunato must know that it is Montresor who is exacting the revenge himself, otherwise, by Montresor's definition, it would not be revenge.

Of course, the fact that the short story never makes it clear precisely what the grevious offence was that plagues Montresor so does beg the question of whether Montresor is entirely reliable - of course, if Fortunato had wronged Montresor so badly it would be unlikely that he would trust Fortunato enough to follow him by himself into the depths of his family catacombs.

Hopefully this will help you work out the theme for other short stories you are studying as well, by using this as an example.

What is the theme of "The Cask of Amontillado," and how does the story embody it?

Unfortunately, you are not allowed to ask more than one question, so I have chosen to focus on the theme of this terrifying short story. From the very first paragraph it is clear that revenge is Montresor's central motive for acting in the way that he does:

The thousand injuries of Fortunato I had borne as best I could; but when he ventured upon insult, I vowed revenge.

Although the exact nature of this "insult" is never divulged, revenge is the driving force that leads Montresor to plot and enact his most gruesome of punishments. Consider Montresor's motto, "Nemo me impune lacessit", which means "Nobody attacks me without punishment", which adds a moment of black humour to the story as Fortunato responds to the firmness of this motto with a strong approbation: '"Good!" he said.' Of course, the first paragraph is key to the reader in terms of displaying Montresor's plan, which is an enactment of his motto:

A wrong is unredressed when retribution overtakes its redresser. It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong.

It is vital therefore to Montresor that Fortunato meets his fate at his hand - Fortunato must know that it is Montresor who is exacting the revenge himself, otherwise, by Montresor's definition, it would not be revenge.

Thus it is revenge that drives Montresor and impels him to devise such a terrible manner of achieving his revenge against Fortunato, which, of course, allows Montresor to observe first hand the revenge he is gaining against his enemy, bricking him in to the depths of his catacombs as he locks away his crime in his mind and re-enters the light of day again.

How can the theme of perversity be analyzed in "The Cask of Amontillado"?

In his short story, "The Cask of Amontillado," Edgar Allan Poe uses the Gothic conventions, but he perverts them by using only humans for the terrible deeds, rather than spiritual forces. It is here that the real horror lies--the capabilities of human beings themselves. 

In addition to his spiritually subversive actions of deceiving Fortunato and luring him into the catacombs so that he can bury him alive, Montresor interacts with Fortunato with perverse mannerisms, almost as though he is seducing him. Near the end of the story, there is much language suggestive of a physical seduction in a sado-masochistic fashion.  For instance, Montresor narrates that Fortunato tries vainly to "pry into the depth of the recess"; then he steps forward and Montresor follows "immediately at his heels." Further, Montresor, in sadistic fashion, fetters Fortunato to the wall. The sexual innuendos in his words are easily apparent,

"pass you hand...over the  wall; you cannot help feeling the niter.  Indeed it is very damp....

"The Amontillado!" ejaculated my friend, not yet recovered from his astonishment.

This is followed by a "low moaning cry from the depth of the recess." This cry is later followed by a sucession of "loud and shrill screams" that Montresor describes as causing him to be "thrust...violently back":

For a brief moment I hesitated--I trembled.  Unsheathing my rapier, I began to grope with it about the recess....But now there came from out the niche a low laugh that erected the hairs upon my head.

This suggestive language serves to subvert the Gothic conventions even further with its perversity. 

The use of sexually suggestive language in this passage indicates how perverse, both psychologically, and possibly physically Montresor is. 

How can the theme of perversity be analyzed in "The Cask of Amontillado"?

In this short story the theme of perversity can be explored through the character of Montresor. The narrative perspective that Poe chooses to use, the first person, is one that allows the reader to see Montresor's perverse mind and the inner-workings of a character who is clearly grappling with sanity. This is something that is signalled in the first paragraph, when Montresor talks of his supposed "injuries":

The thousand injuries of Fortunato I had borne as I best could, but when he ventured upon insult, I vowed revenge. You, who so well know the nature of my soul, will not suppose, however, that I gave utterance to no threat. At length I would be avenged...

As the reader continues the story, there is no evidence whatsoever of any "injuries" that Fortunato had committed against Montresor; indeed, it would be very unlikely for Fortunato to agree to accompany Montresor so easily on such a night if he had actually wronged Montresor so badly. Montresor's determination to gain his revenge and the manner in which he does it is truly shocking and perverse, and presents him almost as an aberration of humanity, and this perversity is something that Poe focuses on very clearly from the first paragraph, which introduces the reader to the workings of a cold-blooded psychopath. In this short story, as in others such as "The Black Cat" and "The Tell-Tale Heart," the narrative perspective allows Poe to introduce the reader to the workings of a madman who is a distinctly unreliable narrator.