Short answers, pulled from the story.
A coif was a simple white linen cap worn by men and women across England and Scotland from the tenth century until the early seventeenth century. This unadorned cap covered the top, back, and sides of the head and served as a practical barrier against the cold and a symbol of modesty.
By the Elizabethan and early Jacobean eras, the humble linen cap transformed into a canvas for intricate blackwork embroidery and delicate lace edging. Wealthy women wore these embellished coifs under gable hoods and various styles of hats to display their social standing to peers.
The Order of the Coif is a prestigious law school honor society founded to recognize academic excellence in law that draws its name directly from the historical headgear. This organization preserves a connection to the past where a now-defunct senior grade of English lawyer known as the Serjeant-at-Law wore a coif even after they were elevated to the position of judge.
For Catholic nuns and religious sisters, the coif was an integral part of the traditional religious habit worn alongside a white cotton cap secured by a bandeau to which the veil was attached. This ensemble included a white wimple or guimpe made of starched linen or cotton designed to cover the cheeks, neck, and chest to emphasize the nun's vow of modesty and devotion.
A mail coif was a formidable piece of armor crafted from interlocking metal rings designed to cover the head, neck, and shoulders while leaving the face exposed for visibility and breathing. This type of headgear was worn by soldiers and knights from the tenth century onward serving as a vital piece of defensive gear in the medieval and early modern periods.
By the early seventeenth century the coif had largely disappeared from the wardrobes of men marking the end of its dominance as a unisex garment. However the coif did not vanish entirely from history as it persisted as an old-fashioned cap for countrywomen and young children preserving a connection to the past in rural communities.