Strictly speaking, the Book of Psalms was the Prayer Book of the Old Testament. It was utilized in prayed and worship for centuries by the Hebrews. When the Christian Church formed (sprouting from Judaism) we continued to use these ancient, time-tested forms of prayer.
The Psalms run like a golden thread woven through the beautiful tapestry that comprises our Orthodox Christian worship. Psalms form the core of Vespers, Matins, and the other daily services such as the Hours, Compline, and Nocturns. Their verses and imagery are found throughout the Divine Liturgy, Marriage, and other Sacraments of the Church, and have provided inspiration for many of the prayers and hymns that we continue to use in our worship to this day. In fact, the Psalms were considered so vitally important that the Early Church decreed that the entire Book of Psalms must be read in its entirety during the course of each and every week; and twice that during the season of Great Lent!
Besides providing inspiration to public worship, the Psalms should also be an indispensable part of our own daily devotions and private rule of prayer. Interestingly, those who sought to pursue monastic orders could not do so until they had fully memorized all 150 Psalms by heart.