Thompson studied at Harvard University and later with the composer Ernest Bloch. He taught at a number of universities and colleges and was director of the Curtis Institute of Music, Philadelphia (1939–41), before joining the faculties at Princeton and, later, Harvard (emeritus professor after 1965).

Thompson's style is conservative and Neoclassical, combining traditional forms with 20th-century styles. It also exhibits a highly developed sense of form and counterpoint.

Of his three symphonies the second was especially successful. Notable among his vocal works are the Alleluia for unaccompanied choir (1940) and The Testament of Freedom for men's voices and orchestra (1942; to words by Thomas Jefferson), both of which achieved great popularity. Thompson's other works include a one-act opera, Solomon and Balkis (1942), and an oratorio, The Nativity According to St. Luke (1965).