Definition and background:
A series of eight services of the Roman Catholic Church performed throughout the day. The divine offices consisted of Psalms, hymns, lessons, responsories, and prayers. Matins 3:00 a.m. Lauds daybreak Prime 6:00 a.m Terce 9:00 a.m Sext noon None 3:00 p.m. Vespers twilight Compline before retiring to bed a series of religious services spread throughout the day that involve prayer, readings and the recitation of psalms. The daily cycle of services consists of Matins, Lauds, Prime, Terce, Sext, None, Vespers, and Compline.
For more information on the music term "Office" check out these other resources:
Wikipedia - Glossary of Musical Terminology
Virginia Tech Multimedia Music Dictionary
ORB -- Medieval Music Glossary