There are two equally important conflicts in this story.
The first conflict is man vs. man as the banker and the lawyer become engrossed in a discussion about capital punishment at a party. This conflict establishes the circumstances for the rest of the story; the two men are each so determined to prove themselves right that they wager two million rubles to prove a point. The banker believes that capital punishment is more "humane," while the lawyer believes that life imprisonment is the better option. The men grow passionately more determined to be correct, and when the banker bets the lawyer that he wouldn't be able to live in solitary confinement for five years, the lawyer increases the terms of the bet to fifteen years—for no additional money. This conflict between the two men creates the context for the second conflict.
As the lawyer spends his years in confinement, both he and the banker face internal conflict. The lawyer studies literature, histories, science, and religion and becomes increasingly disgusted with the society he has left. He ultimately rejects the money he is duly owed, because he no longer finds value in worldly possessions; over fifteen years, his character transforms to a man who sees society as trading "heaven for earth."
The banker, likewise, faces internal conflict. As the years pass, he makes reckless choices in the stock market and loses almost all of his wealth. He can no longer afford to pay the lawyer the two million rubles and even plans to murder him in order to escape the bet and avoid public disgrace.
Both the lawyer and the banker are drastically changed by the end of the story after struggling with their internal conflicts.