Chair arm covers are called antimacassars, according to Dictionary.com. Antimacassars protect the upholstery from wear and soiling, and they add a decorative touch to the chair. They are typically made from stiff crocheted lace, embroidered wool or silk, or the same fabric used on the upholstery of the chair.
Antimacassars are also used to protect the back of an upholstered chair or sofa. The name comes from macassar hair oil, which was popular in the 19th century. Antimacassars protected the back of the piece of furniture from coming in contact with the hair oil of those who were seated on the furniture. In modern times, antimacassars are often simply decorative.