What is the purpose of a Rosary?

When praying the rosary it is also considered a petition. A petition to the Holy Mother to pray for us, we pray for peace, pray for the departed, and pray for others (even those we do not know). I see nothing wrong with having a rosary even though one is not catholic.

[General Introduction to] Devotions to the Blessed Virgin Mary

"'All generations will call me blessed': 'The Church's devotion to the Blessed Virgin is intrinsic to Christian worship.' The Church rightly honors 'the Blessed Virgin with special devotion. From the most ancient times the Blessed Virgin has been honored with the title of "Mother of God," to whose protection the faithful fly in all their dangers and needs .... This very special devotion ... differs essentially from the adoration which is given to the incarnate Word and equally to the Father and the Holy Spirit, and greatly fosters this adoration.' The liturgical feasts dedicated to the Mother of God and Marian prayer, such as the Rosary, an 'epitome of the whole Gospel,' express this devotion to the Virgin Mary." [cf. CCC, 971. The internal quotes are from Lk 1:48; Me 42,56, and LG 66.].

One of these devotions is the Holy Rosary

The Rosary is a centuries-old way of praying. "Medieval piety in the West developed the prayer of the Rosary as a popular substitute for the Liturgy of the Hours." Although obviously Marian in character, it is Christ-centered in its essentials. It is a meditation on the the lives of both our Lord and the Virgin Mary. "Christian prayer tries above all to meditate on the mysteries of Christ, as in ... the Rosary."

"Meditation engages thought, imagination, emotion, and desire. This mobilization of faculties is necessary in order to deepen our convictions of faith, prompt the conversion of our heart, and strengthen our will to follow Christ."

Meditation upon these mysteries leads us to contemplation: "This form of prayerful reflection is of great value, but Christian prayer should go further: to the knowledge of the love of the Lord Jesus, to union with him." [cf. CCC, 2678 & 2708.].

The Rosary has a structure and method designed to help us more readily assimilate the mysteries into our spiritual life. Structurally, it is divided into four parts; each part, into five mysteries. For each mystery is said one "decade" consisting of one Our Father, ten Hail Marys and one Glory Be.

Though at first sight this method of repetition may seem to be tedious and boring, it is better understood when considered as an act of love: "when the Rosary is thought of as an outpouring of that love which tirelessly returns to the person loved with expressions similar in their content but ever fresh in terms of the feeling pervading them." [cf. RVM, 26.].

It is also important to remember that though these repeated invocations are directed to Mary, our acts of love are ultimately directed to Jesus himself, through her intercession. In many Catholic families, there is a pious custom of reciting daily one part of the Rosary.

Source: Devotions to the Blessed Virgin Mary | Handbook of Prayers | Rev. James Socias , Publisher.

The above, (emphasis mine) answer: What is the purpose of a Rosary?

(And as a non-Catholic Christian, should I get one?)

Perhaps this is the way God is calling you to be his son, the son of his Blessed Mother, and brother to her First Born: via love and devotion to his Mother.

I know you will love these incredible stories:

• The remarkable true story of the miraculous intercession of the Virgin Mary in 1944 to prisoner Claude Newman of Mississippi | Mystics of the Church.
• Did Rosary Stop Horrific Serial Killer Ted Bundy? | Spirit Daily.

To pray the Holy Rosary:

• You may get a rosary from here: Sisters of Carmel > Rosary.
• And obtain this pamphlet: Pray the Rosary Daily.

The term "Rosary" refers to a set of prayer beads in a particular configuration, and also to the structured set of prayers which is guided by the beads.

The actual, physical Rosary is a string of beads or knots constructed so as to guide the holder through praying the Rosary prayer. The loop part has 5 sections, separated by larger beads, each containing 10 small beads. Hanging from the loop is a string of 5 beads and a cross.

As the person prays, they start at the Cross and move through the beads up and around the loop. At each bead the appropriate prayer is said. One (of many) online guides to praying the Rosary is here. When moving around the loop, each of the small beads represents a Hail Mary, and the large beads represent a meditation on a particular "mystery" in the life of Jesus or Mary. There are 4 sets of "Mysteries". Some will also use the Rosary structure to pray for other things .. for instance one might pray for one's family on the meditation bead.

Catholics give special significance to the Rosary prayer, for many reasons, including our devotion to Mary. We say it alone, and we say it in groups. Praying a Rosary for the dead is part of our funeral tradition.

We also place strong importance to our actual Rosary beads, which are often among our most prized possessions. I was given a Rosary when I was initiated into the Knights of Columbus. My wife has a Rosary made in the Holy Land that was blessed by the Pope.

I don't think there is anything inappropriate about a non-Catholic having a Rosary or praying the Rosary. If I am wrong I presume I will be corrected!

The Rosary is a very old artifact indeed and has evolved in time, the actual form of the Rosary is not what it was in the very beginning.

Rosary is a style of mental prayer that is based on repetition. This has been a tradition for many centuries since the early ages of monasticism. The emphasis upon repetition of brief, monologistic, prayer formulae laid the foundations for later monastic prayer traditions such as the Jesus Prayer and is the background for analogous forms of devotion in Latin Christianity such as the Hail Mary and Rosary. (Columba Stewart OSB)

The modern form of the Rosary may took its form when Domique de Guzman preached in favor of it. This also led to the story that the Virgin Mary delivered the Rosary to Dominique.

In its actual form, the purpose of the rosary is to meditate in its mysteries while repeating Ave Maria etc… Repetition is useful to achieve concentration and focus. The Mysteries of the Rosary are essential facts in the life of Jesus, except for the Assumption of Mary and the Coronation of the Blessed Virgin which are related to very old church traditions. It is not the purpose of the question to explain all of the mysteries, but a full list can be found in wiki. Meditation in the mysteries of the rosary is something that most christians do but in different ways (reading/studying the Bible, public preaching etc).

Probably you should not get a Rosary. If you want one, or you are curious about this type of prayer I recommend you to read the article or Steward (link above) in order to have more insight in this form of prayer. Then you only need to pray the Rosary and get used to it. Maybe you'll notice how the grace sheds a new light upon the mysteries of our faith.