An entertainment director was an individual responsible for recreational activities.
When Lwaxana Troi welcomed Timicin aboard the USS Enterprise-D in 2367, she told him to think of her as his entertainment director. (TNG: "Half a Life")
When Harry Kim described Marayna to Neelix in 2373, Neelix identified her as one of the entertainment directors of the holographic Paxau Resort. (VOY: "Alter Ego")