The exposition is the reader's introduction to the two main characters. We know the first character as the narrator of the story, but we do not find out that his name is Montresor until much later in the text. The other main character is Fortunato. The most important detail of the exposition is that Fortunato has somehow offended/injured Montresor in a way that cannot be forgiven, so he has vowed to get the ultimate revenge.
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. I must not only punish but punish with impunity.
The complication for Montresor is how to go about getting his revenge and getting away with it. Montresor's plan is to lure Fortunato down into the wine cellar during the carnival festivities one evening under the pretense of sharing a bottle of Amontillado together.
The climax of the story is when Montesor chains Fortunato to the catacomb wall and seals him down there with a brick wall. The entire thing is done while Fortunato is begging for his life.
The resolution is Montresor calmly telling his readers about his deed 50 years later. He also admits that nobody has ever found the body, and that he feels no guilt over the entire thing.