He wrote music for the theater, film, ballet and orchestra, but he is best-known for his operas The Devil and Daniel Webster (1938) and The Ballad of Baby Doe (1956). Moore was born in Cutchogue, New York and died in Greenport Long Island. He went to Yale University, graduating with two degrees in 1917. He served in the Navy as a lieutenant, after which he studied music with Nadia Boulanger, Vincent d'Indy and Ernest Bloch. Apart from classical compositions, Moore also composed several popular songs while at Yale with poet and Hotchkiss School-mate Archibald MacLeish and later in collaboration with John Jacob Niles. These songs were later published in 1921 under the collective title "Songs my Mother never taught Me".
Four Museum Pieces, Moore's first serious work, in its orchestrated version, was first performed by the The Cleveland Orchestra with Moore conducting. It won him a Pulitzer Traveling Fellowship, allowing him to return to Europe to study with Boulanger. In 1926, Moore was invited to join the music faculty at Columbia University, where he remained until his retirement in 1962. He succeeded Daniel Gregory Mason as head of the music department at Columbia University in 1940. He wrote two books on music, Listening to Music (1932) and From Madrigal to modern Music (1942). He also collaborated with fellow Yale alumnus Stephen Vincent Benét on the folk opera The Devil and Daniel Webster (1938).
Moore served as president of the National Institute and American Academy Of Arts And Letters, 1953 - 1956. He had been a member since 1941. In 1954 he was a co-founder, with Otto Luening and Oliver Daniel, of the Composers Recordings Inc. (CRI) record label. He was also the director of the American Society of Composers, Authors and Publishers (ASCAP) from 1957 to 1960.