A Bounty quest is a special quest in which the player must hunt down an outlaw and then hand them in for a reward (normally Kamas).
There are three places where players can start Bounty quests: one in Astrub CityMilitia at (7,-18), one in the Vigilante Base (4,4), one in Amakna CastleMilitia at (5,-6), one in Frigost VillageMilitia at (-76,-42) and one in The Snowbound VillageMilitia at (-75,-74). Each has a different set of targets. At said locations there are a number of posters (at Astrub and The Snowbound Village they are outside while at Amakna Castle and Frigost Village they're inside). If you click on one you can read the wanted notice. Click on the link at the bottom of a scroll and, if your character level is high enough, the quest will be added to your quest log.
Once you have the quest you must find the wanted monster, defeat it in battle, and turn it in to the NPC at the relevant location. If you find two or more bounties in a single mob group they will not all be captured, rather one will be captured though it shows multiple in the loot window.