First of all, in Mark Twain's "The Celebrated Jumping Frog of Calaveras County," the story does not see based heavily upon conflict as many stories often do. If I needed to identify conflicts in the story, the first would be man vs man, in that Jim Smiley is pitted against other men in gambling: for he is a man who will bet on anything. (This also might be considered man vs society, as he bets with everyone in town who will gamble, but man vs man should be sufficient.)
The story that Wheeler imparts starts off sounding like a fairy tall—or a "tall tale," which is a story made up of exaggerations.
There was a feller here once by the name of Jim Smiley…
As interesting a character as Smiley turns out to be, Wheeler is himself a very good storyteller. He continues to paint an interesting and humorous picture of Jim Smiley who is quite the gambler:
[Smiley] was the curiosest man about always betting on any thing that turned up you ever see, if he could get any body to bet on the other side; and if he couldn't, he'd change sides.
In the course of the story, Wheeler introduces various creatures around which Smiley organizes his betting. Several were: Andrew Jackson, his fighting dog; the "fifteen-minute nag" (a horse); and, his frog, named "Dan'l." The winning streak that Smiley has enjoyed finally takes a turn for the worse when he his scammed by a conman who passes through town. This conflict is also man vs man.
The only other conflict may be that of the narrator (Mark Twain) as he tries to make his escape from the company of Simon Wheeler, who tells Twain all about Smiley's escapades. When Twain arrives and asks Wheeler about the Reverend Leonidas W. Smiley (who Twain is looking for)...
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 which follows this paragraph.
It is only at the end, when someone else calls briefly to Wheeler, that Twain narrowly makes his escape.