Praying the Rosary is a very old form of prayer.
As a faithful Jewish man, Jesus would certainly have prayed the Psalms. He quoted from them frequently, even from the Cross. The tradition of praying the psalms daily in private prayer as well as liturgically continued among the first Christians.
At the same time, there arose a love and devotion for the Mother of Jesus as the First Christian, the model of all Christians, and the holiest of God's creation whose body housed the Infant God-made-Man, whose breasts nursed Him, whose love nourished Him, whose wisdom instructed Him, whose devotion allowed her to stand faithfully and sorrowfully in the shadow of the Cross and who received the Holy Spirit as a tongue of fire at Pentecost with the Apostles after Jesus ascended to heaven, and herself taken body and soul into heaven at the end of her earthly life.
The earliest image of Mary, painted nursing the infant Jesus, from the 2nd Century, in the Catacombs of Priscilla, Rome.
Christians have honored Mary from the beginning because she was the Mother of the Savior, Jesus Christ. St Ephrem wrote incredible hymns in honor of Jesus' Mother, and the Saints of the Early Church wrote in praise of her virginity, grace and holiness as Mother of God. Centuries later, Martin Luther, too, wrote in praise and defense of her Perpetual Virginity.
The ancient tradition was kept by the clergy and religious orders of praying all 150 psalms daily in imitation of Jesus and in praise of God the Father. Those that could not read and write, and who could not afford expensive hand-written Psalters (which could cost a year's wages) began the tradition of praying either the Our Father, or the Angel Gabriel's and St Elizabeth's greetings to Mary from the Gospel of Luke, (Hail, full of grace! The Lord is with thee. Blessed art thou among women,and blessed is the fruit of thy womb). A string of knots to accurately count the prayers was often used, creating a chaplet prayer. Later the Hail Mary prayers were divided into groups of ten, and the Our Father prayer, taught by Jesus, was inserted between them.
In the 13th century, St Dominic began to preach against a serious heresy sweeping through the Church. He encouraged the prayer of the chaplet as a Rosarium - a wreath of roses to honor Jesus and His mother Mary - as a way to battle the heresy. St Dominic promoted a chaplet of five decades of "Hail Mary' prayers, to be prayed while meditating on the lives of Jesus and Mary. Many Saints throughout the ages have called the Rosary the best weapon we have against heresy and sin because as we pray in honor of Our Lady, we remember the love of God the Father in sending His only Beloved Son, Jesus, to become man, born of the Virgin Mary and to dwell among us.
The Chaplet of the Rosary consists of twenty decades of Hail Mary prayers (five more were added to the original fifteen by Pope John Paul II), separated by a prayer honoring the Holy Trinity and the Our Father, all prayed while meditating on the lives of Jesus and Mary. Most people do not pray the full twenty decades, but just five decades a day and meditating on one set of Holy Mysteries of the Rosary. The scriptural meditations on the lives of Jesus and His Mother are divided into four sets of five Mysteries, reflecting different themes, Joyful Mysteries (the early life of Jesus), Luminous Mysteries (Jesus' early ministry), Sorrowful Mysteries (Jesus' death) and Glorious Mysteries (Jesus' Resurrection and events after). The prayers are to be said with reverence while meditating on one Mystery per decade.
You can read more about this wonderful, meditative prayer honoring Jesus and His Mother here and here.
The Rosary prayed with the Scriptures.
The prayers and Mysteries of the Rosary, and how to pray it properly.
Information on devotion to Mary and the Saints, and why Catholics rightly pray as we do here.
"Thou alone, O Lord, and Thy mother are they who in every respect are wholly beautiful; for there is no spot in Thee, O Lord, nor any stain in Thy mother."
--St Ephrem The Syrian, 4th Century AD