What do you call to a person who writes a play?

Playwright is a person who writes a play. A playwright writes the script of the play. Sometimes playwright can be called as dramatist.

Just as a poet writes poems, a playwright writes plays. If the spelling of playwright looks odd, that's because wright is a word for a craftsperson or someone who builds things (like a shipwright builds ships). Given the juggling of characters, scenes, and plots in a play, you can see how writing a play is like building something. The most famous playwright ever in English is William Shakespeare.