Epessë is a nickname or honorific choosen by an elf as his or her public name later in life. It is part of the essë, or naming, of High Elves, along with the essi ("father-name") and amilessë ("mother-name").
An example is Gil-galad ("Star of Radiance"), which was the mother-name of the last High King of the Noldor. His father-name was Rodnor. Upon becoming king he was given the epessë Ereinion ("Scion of Kings") and called Ereinion Gil-galad.