Q: How did the Rosary originate?
A: While examining the life of St. Dominic (died 1221) we can find the first definite reference to this form of prayer. The Saint taught and promoted the Rosary in France in response to a heresy which was endangering the people's faith. This method of prayer was adopted by the Dominicans and they are still its most constant supporters. But, it is highly probable (and often taught) that it originated earlier than that, as a way for the lay people to participate with the Divine Office, through which the monks recited the 150 Psalms throughout the course of the day. Since the majority of people could not read at the time, they would instead say anywhere from 50 to 150 Ave Marias (Or "Hail Mary's"). Some began using ropes with knots to keep track of the number of prayers said.
Q: What does "Rosary" mean?
A: Rosary literally means "crown of roses", in that it is a spiritual bouquet which is presented to Our Lady.
Q: Why should Catholics pray the Rosary, and is it required?
A: The Rosary has been promoted and highly encouraged by countless Saints and Popes. Our Lady herself repeatedly counsels the faithful to use this form of prayer during her apparitions, particularly at Fatima. Many indulgences and spiritual promises are attached to it, and it is often prescribed as a form of penance after Confession. It is a wonderful way in which to meditate upon Scripture while contemplating Christ's life and great virtues, and learning how to apply His example to your own life. It is also a comforting way to offer up a particular intention to Mary, who is always at Christ's side. That being said, the Rosary is only required if assigned by a priest as penance. Otherwise, Catholics are not required to say the Rosary, and it is okay if they prefer a different form of prayer.
Q: How should the Rosary be said?
A: To be technical, the Rosary is not "said", but "prayed". Though praying it aloud in groups is fine, it is recommended that it be said slowly and quietly, so that one might meditate upon the mysteries, and to avoid the repetitious prayers from becoming words without meaning. Pope Pius VI said this about how to pray the Rosary:
There has also been felt with greater urgency the need to point out once more the importance of a further essential element in the rosary, in addition to the value of the elements of praise and petition, namely the element of contemplation. Without this the rosary is a body without a soul, and its recitation is in danger of becoming a mechanical repetition of formulas and of going counter to the warning of Christ: "And in praying do not heap up empty phrases as the Gentiles do; for they think that they will be heard for their many words" (Mt 6:7). By its nature the recitation of the rosary calls for a quiet rhythm and a lingering pace, helping the individual to meditate on the mysteries of the Lord's life as seen through the eyes of her who was closest to the Lord. In this way the unfathomable riches of these mysteries are unfolded. (Marialis Cultus 47).Each day has a set of mysteries which are recommended, but not required. When praying it privately, there is room for one to customize it as necessary, as long as the aim is still to spend time prayerfully reflecting upon Christ, asking Mary's assistance in doing so, and asking for Heaven's assistance regarding your personal intentions.
I'll leave you with a quote from Bishop Hugh Doyle, which sums up why the Rosary could greatly increase and aid your spiritual journey:
"No one can live continually in sin and continue to pray the Rosary: either they will give up sin or they will give up the Rosary."
May God bless you all today!
Information found at EWTN and Catholic Answers.