Name of scarf worn by priests ? – Religion & SpiritualityReport
It is called a stole and is worn by deacons, priests, and bishops alike. However, the way they wear it is different. Priests and deacons wear it in the same way – in the “scarf-like” fashion you mentioned. Conversely, deacons wear it as a sash, going around their left shoulder, chest, and back. It signifies authority in the Church, as priests, deacons, and bishops guide and teach us about Christ and the Church.
“[…]the stole is only used, and must be used, at a function peculiar to the deacon, priest, and bishop, a function that presupposes the order (e.g., at the celebration of Mass, when the Blessed Sacrament is touched, when the sacraments are administered), but not for example, in processions or at Vespers” (link added). However, a bishop may wear a stole during Vespers. He is an exception.