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!