a hard, square, ceremonial hat with three or four vertical projections and sometimes with a pompom or tassel at the top, worn by Roman Catholic clergy
See biretta in American Heritage Dictionary 4
also be·ret·ta or ber·ret·ta(bə-rĕtˈə)
noun Ecclesiastical
A stiff square cap with three or four ridges across the crown. Birettas are worn especially by Roman Catholic clergy and are black for priests, purple for bishops, and red for cardinals.