Each player has 10 bounty slots that they may fill at any one time - this can be done here. You may place a bounty on any one you choose, for any amount you like. To place a bounty a 50% listing fee is added, and if you wish to remain anonymous and further 50% of the initial bounty amount is also added (for example, an anonymous $10 bounty will cost you $20). A player may have as many bounties as possible put upon them if many people decide to group and target them, however unclaimed bounties will expire after 7 days. Bounties can not be bought off or removed via any other means.
Bounties can be claimed whether you are in Torn or abroad, all you have to do is hit "attack" via their profile or "claim" through the bounty page. You must hospitalise your enemy to claim a bounty - and you may no longer claim a bounty you have set on someone. There are 5 awards available for collecting bounties: collecting 25, 100 and 500 will give you 3 medals, collecting 250 bounties and collecting $10,000,000 in bounties will give you 2 honors.
If you have an anonymous bounty placed on you, anyone in a 7* Cyber Cafe can reveal the bountier for you - although anyone with the SYS stock benefit is protected from this!
Take caution when looking for bounties to claim - some strong players may use them to farm defends from unsuspecting noobs! Be sure to check out their profile and personal stats to make sure you can win the fight.
All of this was changed with the bounty update on 14/02/17.