There are two types of bounty hunting. The first is accomplished simply by destroying enemy ships, which will gain you a profit depending on the relations of your alliance and the other ship's, as well as the type of ship destroyed. In wartime, the bounty from destroying enemy ships is often enough to fund your personal war effort, with the possibility of a bit of profit on the side.
The second type of bounty hunting involves capturing people wanted for certain offenses. As of this writing, the only offense you're likely to capture someone for is transporting contraband. If you see a ship containing contraband, you can TAG it for a bounty hunt. You then have 24 hours to capture the ship's owner and turn him or her into a security office in the same area of space where the offense was committed, which can be found on any capital planet. Simply stun the pilot, go to a security office, and ARREST him or her to collect your bounty.