This traditional pleated covering is commonly worn today by women in various Amish, Mennonite, Brethren, Old Order groups. It is called a kapp, cap, headcovering, prayer covering, or veiling.
Different groups will have different variations, this is a commonly worn style. The full crown is pleated to fit a squared brim. White ribbon or organdy ties are sewn to the brim corners. Ladies will tie them beneath the chin, or just leave to hang untied. In some groups, the women wear black ribbon ties on their covering when married, white for unmarried girls. The devotional covering is worn daily, often covered with a black outing bonnet when traveling away from home. This sewing pattern for your own Amish headcovering includes sewing instructions and patterns for multiple ladies and girls sizes. This design is from the Dingman family at Friends Patterns.