A storyteller was someone who read or recounted stories for the enjoyment of others.
The Sirah was the storyteller of a Faren Kag's village on Bajor. It was the Sirah's role to frighten away a mythical energy creature known as the Dal'Rok, which appeared for five days each year, by telling a story of the village's creation. In 2369, Chief Miles O'Brien was briefly appointed the Sirah, before Hovath took over the role. (DS9: "The Storyteller")
The Doctor didn't consider himself to be a very good storyteller either, something he admitted to a hologram of Hrothgar in 2371, after the king asked him to tell a story of his home. (VOY: "Heroes and Demons")
Neelix was known for telling a story on occasion. In 2377, while supervising the recently freed Borg children Icheb, Mezoti, Azan and Rebi, he made up a story to explain the apparent power-loss the USS Voyager was experiencing at the time. In an effort to divert the childrens' attention, Neelix embarked upon a ghost story involving the ship's deck twelve. (VOY: "The Haunting of Deck Twelve")