A mitre is a tall hat worn by leaders in some Christian churches. In the Roman Catholic church, it is worn by bishops, archbishops or cardinals (higher order). The front and back are the shape of a triangle. It is only worn when the bishop is dressed in his special religious clothes, but is used on a coat of arms to show that the owner is a bishop.
Mitres are also worn by some clergy in the Eastern Orthodox and Eastern Catholic churches.