What were the terms of the bet in "The Bet"?

If the lawyer could stay in solitary confinement for fifteen years, the banker would give him two million rubles as prize money.

According to the bet, the lawyer would have to spend the fifteen years of his imprisonment “under the strictest supervision” in a lodge situated in the banker’s garden. During this period, he couldn't step beyond the doorstep of the lodge nor meet or see any human beings. He couldn't even hear any human voice or “receive letters” from friends and relatives. Even newspapers would be denied to him.

Nevertheless, the agreement permitted the lawyer to order anything else he wanted, including books, music and wine, “in any quantity he desired.” His solitary confinement would begin at twelve o'clock on November 14, 1870 and end at twelve o'clock on November 14, 1885. If the lawyer breached any of the terms of the bet even in the slightest degree at any moment during the agreed time period of the bet, the banker would be under no obligation to pay him two million rubles.

There was a party. Guests at the party discussed capital punishment. A banker argued with a lawyer. The banker argued that capital punishment was more humane than life imprisonment. The lawyer argued that he would choose life imprisonment. The banker and the lawyer debated. The debate became more heated. The two began to argue.

Finally, the banker struck up a bet with the lawyer. The banker bet two million rubles that the lawyer could not stay imprisoned for fifteen years. The lawyer insisted that he could stay imprisoned for fifteen years. The lawyer accepted the banker's challenge. They agreed that the banker would pay the lawyer two million rubles at the end of the lawyer's imprisonment.

In the end, the lawyer walked away from the bet without taking the banker's money. During his imprisonment, he changed. He learned what the meaning of life is. Life is not about money.   

What circumstances led to the bet in "The Bet" and what were its terms?

The subject of the bet arose in the context of of a party, during which there was conversation concerning the morality of the death penalty and whether it ought to be replaced by life imprisonment. In these conversations, the banker voices his opinion that execution is the preferable punishment, as life imprisonment will drag on over time, continually sapping the spirits of the condemned. The lawyer disagrees, holding that at least the prisoner will retain his life, and on this grounds, life imprisonment is preferable to death.

The banker is incensed with the lawyer's challenge. He responds by throwing down a gauntlet. He offers to pay the lawyer two million rubles if he can stay imprisoned for a span of five years. The lawyer's response is to up the ante, by claiming he can remain a prisoner for fifteen years' time. The banker agrees, and the terms are laid out.

The lawyer would remain imprisoned for a span of fifteen years. He would be imprisoned in a wing of the banker's house and forbidden from having any human contact with the outside world. However, he would have access to books, he'd be allowed to play a musical instrument, he'd be allowed to write letters, and he'd have access to wine and tobacco. Should the lawyer break this agreement or leave his imprisonment, then the bet would be void, and the banker would not be required to pay him the money.