A playwright, or dramatist is a person who writes plays.
On Earth one of the most famous playwrights was the 16th century human William Shakespeare. Both former Governor Kodos of Tarsus IV and Starfleet Captain Jean-Luc Picard owned copies of the complete works of Shakespeare. (DSC novel: Drastic Measures; TNG episode: "Hide and Q")
Two other famous playwrights from Earth were the 19th century humans Gilbert and Sullivan. (TNG episode: "Disaster"; TNG movie: Insurrection)
In the Delta Quadrant, Kelis was a playwright on an L-class planet. Working with B'Elanna Torres he created a play based on the adventures of the USS Voyager. (VOY episode: "Muse")