Ballet was a graceful form of dance with accompanying music, and also the name given to a type of performance that involved ballet. Female ballet dancers were known as ballerinas.
Some ballet performances from Earth were Coppélia (DS9 novel: Hollow Men) and The Dying Swan (VOY episode: "Coda"). The Orions also developed ballets during their New Days (FASA RPG module: The Orions: Book of Common Knowledge).
Kathryn Janeway took ballet classes and learned The Dying Swan when she was six years old (VOY episode: "Coda"). Julian Bashir dated a ballerina named Palis Delon when he was at Starfleet Academy (DS9 episode: "Armageddon Game"). In 2364, when the USS Enterprise-D traveled hurtled into deep space, a hallucinating crewwoman took up ballet dancing (TNG episode: "Where No One Has Gone Before").