misisons for hollows may involve hunts for souls, hunts for hollows, even hunts for shinigami. you may be required to assault locations known to have high spiritual awareness souls, you may be sent to gather information. you may go to hueco mundo, the world of the living, and sometimes even soul society.
a mission may be given to you by any hollow of a rank higher than you, apart from a gillian, so adjucha can order hollows around, and arrancar can order adjucha about. missions normally can't be refused.
if a member of a fraccion gives you a mission it will be under their fraccion number, same for espada, but if an ordinary hollow gives you it, they can place it under ay fraccion number, and they will tell you which number it was.
admins can also give you misisons.