The major conflict in the story occurs inside the daughter of the king. Her beloved is being forced to undergo a barbaric "trial" in which he must pick a door in the arena and either be eaten by a tiger or married to a beautiful woman who is a rival to the princess.
This leads to the princess being torn between altruism and jealousy, which becomes the first conflict. The same powerful emotion of love that has driven her to find out what door the tiger is behind will lead her to anguish if her beloved dies but also to anguish if she has to watch him every day married to another woman. She has to decide if saving his life is worth the pain it will cost her.
Second, the story sets up a conflict between superstition and rationalism. The whole idea of setting up a system of reward and punishment based on "poetic justice"—the idea that the fates or providence will give out to people their just reward—is highly questionable, and the story invites us to debate the king's methods.
Third, the story—as Stockton intended—sets up the conflict inside of us as to what we would do if we were in the princess's position. All of us might think we would take the altruistic path, but often people serve self-interest. The story is a vehicle for exploring why we might or might not save the person in question.