Basic Information
A coif is a hood of flexible armour, similar to a modern balaclava, worn to protect the head, neck and face. Like a modern balaclava, the coif could have a variety of openings for the face, ranging from a full sized hole (normal) to mere eye-holes (rare). A coif could be worn under a helmet or instead of one.
Common choices for coif-material would be padded armour and mail (frequently with the mail atop the padding), but coifs could also be made of leather or even plant material (such as the "rope helmets" found in parts of the Pacific Rim).
Modern coifs - which tend not be called coifs - are often worn for protection against flash (or flames generally), but may include slash resistant fibres to ward off knife attacks.
Note that a "coif" - usually of mail - that only protects the neck and shoulders, rather than covering the head, is known as a pixanne.