An essay was a writing assignment on any given subject written by an individual with a particular point of view.
In 2369, the students in Keiko O'Brien's class were given the assignment of writing an essay about ethics. However, Nog claimed to the substitute teacher, Miles O'Brien, that his essay was stolen by Vulcans in Quark's. (DS9: "The Nagus")
In 2375, The Doctor had created at least two essays, one holographic, entitled "Under the Skin" (VOY: "Nothing Human") and one photographic essay called "A Day in the Life of the Warp Core". (VOY: "Juggernaut")
In 2378, Captain Kathryn Janeway assigned Icheb to write an essay detailing Federation history as part of his training for Starfleet Academy. She later had Q's son, Q, write an essay of his own about the history of the Q Continuum which he called "I, Q: An Insider's View of the Continuum". (VOY: "Q2")