Bounties last for 7 days, plus an extra grace day. If you don't assign the bounty to any of the existing answers, you don't accept any of the answers*, and there isn't any answer to which the bounty can be automatically assigned, the bounty is simply removed. In any case, the bounty ends; if you don't accept an answer, or assign the bounty before it ends, you cannot anymore assign it.
You are notified when the bounty expires within three days, or in the next 24 hours. The notifications are shown in the "notifications" tab of the Stack Exchange drop-down menu, the one visible to the left top side of the screen. It is the same drop-down menu that notifies you about new answers to your questions, or new comments.
* This is in the case you offered a bounty for your own question. When you offer a bounty for a question that is not asked by you, you cannot accept an answer.