I assume you're talking about the Orarion, which is what the Deacons in the Eastern (Byzantine) Orthodox tradition wear, over their left shoulder. The similarity with "oorar" in Armenian makes me think we're talking about the same thing.
Ὁράριον is the Greek term, but it's all I've heard used in English. It actually comes from the Latin verb "oro", which means "to pray, supplicate, entreat." So it means, more or less, "the prayer thing," or, if you will, the "prayer scarf/band/strap." Hence why the deacon holds it aloft as he intones the petitions, as it represents his particular office in voicing the prayers of the people.