He studied pianoforte with Antoine François Marmontel and composition with César Franck.
In 1897 Bordes published Archives de la tradition basque, an ethnomusicological study commissioned by the French minister of public education.
Bordes founded the Schola Cantorum, a society for sacred music, with Vincent d'Indy and Alexandre Guilmant.
The Schola Cantorum was responsible for reviving interest in plain-song and the music of Palestrina, Josquin des Prez Victoria and others.
He remained actively involved with the original Schola Cantorum until his early death at Toulon.