The climax of a story is usually defined as the point of the greatest intensity of the story; it is often referred to as the story's turning point or crisis. It is when the direction of the plot turns dramatically in a different direction.
It is also possible for different people to identify the climax of a story in different places. In To Kill a Mockingbird, some might see Tom Robinson's death as the story's climax as so much of the novel centers around his trial. However, I would look to Bob Ewell's attack on the children as the story's turning point. Surprisingly, as with "The Most Dangerous Game," written by Richard Edward Connell, the climax of the story can take place where you would not expect it: in this short story, it occurs almost at the very end of the story. In other pieces of literature, there may be more falling action before the story's resolution.
In Edgar Allan Poe's "The Cask of Amontillado," I would say that the climax occurs when Montresor places Fortunato within the enclosure he has fashioned, and shackles him to the wall, but not ending until Fortunato stops screaming. For me, Fortunato's hysterical response is the point of highest intensity in the story: it makes me empathetic for the man who unknowingly has stepped into the trap his "seeming" friend Montresor has prepared for him. The reader is left to wonder (in light of Montresor's apparent madness) if Fortunato ever really did anything to him, or if Montresor imagined it: for the narrator never does reveal the "insult" for which he punishes Fortunato, and Poe creates his character as one on the brink of madness (if not already "swimming in it"). It is Fortunato's last scream that signals the end of the climax and the beginning of the falling action.
I replied to the yells of him who clamored. I re-echoed—I aided—I surpassed them in volume and in strength. I did this, and the clamorer grew still.
The entire tale has been about Montresor luring Fortunato into the catacombs with the intent to murder him. It is when that objective is reached—and when Fortunato realizes what is happening to him— that I see the direction of the story changing from "plan" to "mission accomplished." From here, the plot moves to the falling action and the resolution.