At the tower there's a robut who hands out the bounties. You have to choose the ones you want to accept, as you can only have up to 5 bounties active at any time. They show up in your inventory in the bottom right, and get a little gold star when completed. Then you return them to the bounty bot for xp n stuffs

Pick them up from the Xander in the main courtyard. They go into your inventory and there they stay until you've fullfilled the requirements (kill 100 enemies without dying for example), after you do, go back and tell him for your rewards.