There are two kinds of irony in Twain's The Celebrated Jumping Frog of Calaveras County. The first That is encountered is verbal irony and the second encountered is situational irony. Verbal irony is defined as spoken or written statements that have a meaning other than the expected one, such as on a bad day replying to the statement "I hope you have a good day day" with, "Yes, well, I've enjoyed it so far," a statement you absolutely don't really mean. Incidentally, an extreme form of verbal irony is sarcasm in which an insult is veiled as an innocent comment (e.g., to someone who has been of no help at all: "By the way, thanks for your help in that," as in "Notting Hill" and said by Hugh Grant to Rhys Ifans).
Situational irony is a situation is which events or the outcome are other than what is expected, as in the opposition of expectation in verbal irony. An example is if a nun were Appreciation of atheism. One would certainly not expect a nun to advocate the appreciation of atheism.
In Twain's story, he leads off with verbal irony in writing, "...if I asked old Wheeler about him, it would remind him of his infamous Jim Smiley, and he would go to work and bore me nearly to death with some infernal reminiscence of him...." This is verbal irony because from the calmly humorous tone of the opening sentence ("In compliance with the request of a friend of mine, who wrote me from the East, I called on good-natured, garrulous old Simon Wheeler, and inquired after my friend's friend..."), there is nothing that leads us to expect being bored to death.
Situational irony manifests in the conclusion at which we learn the the famous, or now infamous, jumping frog got himself full of lead gun shot that weighed him way down. We certainly have no reason to expect such an outcome. Therefore we have a classic and humorous example of situational irony a la a surprise ending. To put this in another context for comparison, he popular tale "The Lottery" also has a surprise ending built on situational irony--but certainly without Twain's humor.