The bounty system is a way for users with over 75 rep to provide a 50-500 rep incentive to answer a question that's at least 2 days old.
Bounties expire after seven days. After a bounty expires, there is an additional 24 hour window for the bounty to be awarded so that late answers may be considered. If it remains unawarded and one or more answers posted during the bounty period have a score of 2 or more, half the bounty will be automatically awarded to the one with the highest score.
When offering a bounty, you choose one of the options from the reasons why you are starting the bounty (e.g., draw attention, authoritative reference needed, reward existing answer, etc.) And you can also add a text with more detailed explanation.