The term for a movie's first appearance or a girl's formal presentation to society comes from the French debut, meaning "from the mark." That's because, originally, debut was a term for the opening move in bowling, billiards, and archery - the first shot at the target, the first stroke, the first die cast, etc. (In French, de means "of" and but means "butt or target.") The term is similar to "your move" in chess and "your play" in cards. The sense of "to lead off in play" was eventually extended to mean to start out at anything - a new career or a hot car or a bold fashion.
SOURCES: 'The Encyclopedia of Word and Phrase Origins,' by Robert Hendrickson; 'The Barnhart Dictionary of Etymology,' by Robert K. Barnhart; 'Etymological Dictionary of the English Language,' by Walter W. Skeat.