Final answer:
Leonard Bernstein is the composer of 'West Side Story' and 'On the Town,' known for his contributions to American musical theater and his fame as a conductor.
Step-by-step explanation:
The composer of the musicals "West Side Story" and "On the Town" who also became one of the most outstanding conductors in American history was Leonard Bernstein. His work in American musical theater is legendary, particularly with his groundbreaking approach in West Side Story, where he, alongside his co-creators Jerome Robbins (original director/choreographer), Arthur Laurents (book), and a young Stephen Sondheim (lyricist), reimagined Shakespeare's Romeo and Juliet. The work was a reflection of the time's social issues, portraying the story of feuding Puerto Rican and white gangs in New York City. Bernstein's contributions to music extend beyond composition; his tenure as a conductor brought classical music to the masses through his educational concerts and recordings, solidifying his place in American cultural history.