It is usually suggested that the rosary began as a practice by the laity to imitate the monastic Office (Breviary or Liturgy of the Hours), by which monks prayed the 150 Psalms. The laity, many of whom could not read, substituted 50 or 150 Ave Marias for the Psalms.
Sometimes a cord with counters on it was used to keep an accurate count.
The first clear historical reference to the rosary, however, is from the life of St. Dominic, the founder of the Order of Preachers or Dominicans. He preached a form of the rosary in France at the time that the Albigensian heresy was devastating the faith there. Tradition has it that the Blessed Mother herself asked for the practice as an antidote for heresy and sin.
One of Dominic's future disciples, Alain de Roche, began to establish Rosary Confraternities to promote the praying of the rosary. The form of the rosary we have today is believed to date from his time. Over the centuries the saints and popes have highly recommended the rosary, the greatest prayer in the Church after the Mass and Liturgy of the Hours. Not surprisingly, it's most active promoters have been Dominicans.
Rosary means a crown of roses, a spiritual bouquet given to the Blessed Mother.
It is sometimes called the Dominican Rosary, to distinguish it from other rosary-like prayers (e.g. Franciscan Rosary of the Seven Joys, Servite Rosary of the Seven Sorrows). It is also, in a general sense, a form of chaplet or corona (also referring to a crown), of which there are many varieties in the Church.
Finally, in English it has been called "Our Lady's Psalter" or "the beads."
This last derives from an Old English word for prayers (bede) and to request (biddan or bid).