You know what, I grew up Catholic, and I didn't even know what the heck the Daily Office was until I began visiting Anglican parishes!
The Divine Office, or Daily Office, is a very traditional ritual prayer in the Christian tradition. The early monastics would dedicate certain times of the day for prayer - filled with prayers of penitence, Bible passages, and prayers written by ancient Christians, along with standing and kneeling at different times. These became quite formalised where almost every waking hour was dedicated to this format of prayer!
Eventually it became quite complicated and privy only to the monastics, and the laypeople developed alternative devotions instead, such as the Angelus or the Rosary.
Thomas Cranmer, who saw this problem, translated and simplified the Daily Office from its matins, mid-day prayer, vespers, compline, sexts, nones, etc. into Morning Prayer and Evening Prayer in the language understood by the people. This eventually became the Book of Common Prayer.
Many of the prayers in the BCP are translations of ancient Christian prayers and a few penned by Cranmer himself. By praying the Daily Office, you are essentially praying with the monastics and clergypeople both ancient and modern, the Orthodox, the Catholic, the traditional Lutheran, and the Anglican. You are reciting, genuflecting and standing to the rhythms of the Divine Liturgy that has withstood for a thousand years.
Good luck! Most lay Catholics do not do the Divine Office, as it is complicated, but the Anglicans are blessed that they have it at the forefront of their faith. I do not pray it all the time, but once in a while, I absolutely love it!
God bless!