Irony is a difficult concept to precisely nail down, partly because the word has been so loosely applied in common usage, and partly because its definitions overlap with similar but distinct concepts like sarcasm.
Verbal irony might best be defined as saying something that contrasts with what is meant, or with observable reality. There is also an element of intention, either by the speaker or, in the case of literature, by the author, for the contradictory nature of the expression to be apparent to the audience.
One example in The Cask of Amontillado, and probably the most obvious, is Montresor's cruel "Yes, for the love of God!". This has multiple meanings, such as indicating that Montresor believes his actions are righteous, or that he is mocking Fortunato (as in "yes, yes, blah blah for the love of God, I'm enjoying this"). The irony in this quote is in its implications of Godly love; what Montresor is doing is anything but loving or Godly, and there is no interpretation in which this does not strike the reader as the opposite of the meaning of the words.
Another example might include the following exchange:
"Enough," he said; "the cough is a mere nothing; it will not kill me. I shall not die of a cough."
"True -- true," I replied-
In this case, we can forgive Fortunato for not recognizing Montresor's foreshadowing for the threat that it is; Montresor merely seems to be agreeing and supportive, but of what point is uncertain. Maybe he's agreeing that the cough is a "mere nothing", or he's just hurrying Fortunato along; his words seem innocuous. What is clearly meant, at least to the reader, and only because we know of Montresor's intentions, is that Montresor is saying "It's true that you won't die of a cough; you'll die from being bricked up inside these catacombs."