A shinigami mission will normally involve exterminating hollows, although sometimes it may involve going to certain places and setting up devices/extracting data, you may even be required to exterminate fellow shinigami, or preform other duties.
your mission will be given to you by your captain, or an admin. Lieutenants may also hand missions out to their squad members. although, as you get higher in level, you may get less missions, as there will be less things suitable to you.
if you want a mission ask an admin or your captain, if they really want to, (they probably won't.) they can refuse your request.
