All three terms—Breviary, Liturgy of the Hours and Divine Office—mean the same thing.
Basically, it is a cycle of daily prayer that is celebrated by clerics and religious, and it is part of the public liturgy of the Church, along with the celebration of the Sacraments and Benediction of the Most Blessed Sacrament. It is designed to sanctify the hours of the day.
Divine Office connotes sacred duty or opus Dei, the work of God. Since the Second Vatican Council, the laity has been encouraged to pray at least parts of the Divine Office. Many parishes celebrate either Morning or Evening Prayer in Church. It has been traditional since even before the Council to solemnly pray Vespers or Evening Prayer, especially on Sunday evening.
The Catechism of the Catholic Church states in #1175, “The Liturgy of the Hours is intended to become the prayer of the whole People of God. In it Christ Himself ‘continues His priestly work through His Church.’ His members participate according to their own place in the Church and the circumstances of their lives: priests devoted to the pastoral ministry, because they are called to remain diligent in prayer and the service of the word; religious, by the charism of their consecrated lives; all the faithful as much as possible.”
The Liturgy of the Hours is divided up into seven hours or times which are as follows: the Office of Readings, Morning Prayer, Midmorning Prayer, Midday Prayer, Daytime Prayer, Midafternoon Prayer, Evening Prayer, and Night Prayer. There are common elements to the different prayers: introduction, hymn, psalms, reading, responsory, intercessions, Lord’s prayer, final prayer, and conclusion. In Morning Prayer, after the responsory, the Canticle of Zechariah is prayed. At Evening Prayer, after the responsory, the Canticle of Mary is prayed, while at Night Prayer the Canticle of Simeon is said.
The Divine Office covers the span of a year. In addition to the psalms and scripture readings it follows the liturgical seasons of the year: Advent-Christmas, Lent-Easter, and Ordinary Time. Throughout the year, saint and martyr days are observed, as well as feasts of Jesus and Mary.
The Little Office of the Blessed Virgin Mary, which was introduced by Saint Peter Damian in the eleventh century, is a shorter version of the Liturgy of the Hours that is traditionally used by many religious communities and members of sodalities.