An active bounty is a bounty that is not yet awarded and has not yet expired.
When a user wants to draw attention to a question, he/she can offer a bounty - offer a piece of his/her own reputation.
A bounty is active until the user awards it. If a bounty is not awarded within 7 days (with a 24 hour grace period), the bounty automatically expires and is no longer active.
It is possible for a bounty to be automatically awarded after it expires. Half the bounty value is awarded to the highest scoring answer written during the bounty period, provided it has a score of 2+ and was not written by the one who offered the bounty.
For all details about the bounty system, see: How does the bounty system work?