The Liturgy of the Hours, or Divine Office, is "the public prayer of the Church", in which we hear the word of God and pray the psalms throughout the day. DivineOffice.org presents a very thorough set of General Instructions online. The Catechism teaches on the Liturgy of the Hours in paragraphs 1174-1178.
In part, the Catechism tells us that:
- "The Liturgy of the Hours is intended to become the prayer of the whole People of God." 
- "The celebration of the Liturgy of the Hours demands not only harmonizing the voice with the praying heart, but also a deeper understanding of the liturgy and of the Bible, especially of the Psalms." 
- "The hymns and litanies of the Liturgy of the Hours integrate the prayer of the psalms into the age of the Church, expressing the symbolism of the time of day, the liturgical season, or the feast being celebrated." 
Who prays the Divine Office?
The Holy Father explains that "Bishops, priests and deacons aspiring to the priesthood, all of whom have been charged by the Church to celebrate this liturgy, are obliged to pray \ all the Hours daily." What about the rest of us? Pope Benedict tells us that "(t)he Synod asked that this prayer become more widespread among the People of God, particularly the recitation of Morning Prayer and Evening Prayer." He recommends " that, wherever possible, parishes and religious communities promote this prayer with the participation of the lay faithful."
What if there isn't a local group praying the Divine Office? You can join in the public prayer of the Church on your own. It is public prayer, after all - jump right in. A lay person may join this worldwide prayer at any time, praying as many or few of the hours as desired.
How do I pray the Divine Office?
Traditionally, the Liturgy of the Hours has been set in a series of books. You can purchase a full four-volume set (Liturgy of the Hours (4-Volume Set), or Large Print) or a condensed single volume (Christian Prayer : The Liturgy of the Hours). You can also find text and audio versions at several websites, like universalis.com and divineoffice.org.
If you do want to use the printed book(s), it will help to have a guide. The classic English guide to praying the Divine Office is the St. Joseph Guide. You can find the 2012 version by searching for "Saint Joseph Guide for Christian Prayer 2012".
Free Trial - Satisfaction Guaranteed
The online versions can serve as a "free trial" (no commitment, satisfaction guaranteed; though I can't promise an operator won't call you) before you buy books or as your sole source of the day's prayers.
Try a single hour and see how you like it. You can pray the morning hour before you leave for work. If, like me, you're not a "morning person", you can try praying the evening hour before you leave work (or before your spouse gets home); it provides a clear transition between your day and your evening. You can also pray the night hour right before bed.