In what way does Twain uses irony in "The Celebrated Jumping Frog of Calaveras County"?
Of course, irony is one of the principal tools in Twain's kit that he uses to achieve the immensely comic effects in his stories. In this traditional "tall tale," irony is used in the framing narrative to set up the central narrative, the tale of the frog, and to present the narrator that Twain is so enraptured by. Note how he introduces Simon Wheeler in the second paragraph of this text:
Simon Wheeler backed me into a corner and blockaded me there with his chair and then sat me down and reeled off the monotonous narrative that follows this paragraph.
Twain, being a humorist, is unlikely to relate to us a "monotonous narrative" that he goes on to call "interminable" just a sentence or two later. This is an example of verbal irony, as Twain is being very economical with the truth. As we read on, we discover that the story that follows is anything but "monotonous."
You might like to consider the irony within the main story itself, when Smiley, the consummate trickster, is himself tricked in the most absurd of ways: by filling his frog with quail shot. In addition, there is dramatic irony as we know what has happened to the frog whereas Smiley does not.