A bounty is an in game challenge that requires you to eliminate a particular player during the match. A rough location of where the player is is marked on your map. The player who is the bounty will also be notified that they are being hunted and they will bbe given a indicator of how close you are to them

You can get a bounty by using one of the bounty board or talking to specific NPCs that offer bounties for sale.

This map shows all the locations where you can get a bounty.

Map credit to fortnite.gg