As a leadership title (along with chairman, chairwoman, chairperson).
A faculty member who is appointed to a named professorship is said to hold or occupy a chair. Any chair (any named professorship) can be called an endowed chair, because it is supported by an endowment (funds specifically set aside or donated for the purpose, from either internal or external sources).
Named professorships are capitalized. Often the full title includes first names and middle initials of the donors; they may be omitted in normal use.
On second reference, the field may often be omitted as well. If Jane Doe is the Beatrice E. Dante Professor of Medieval Imagery, she holds the Beatrice E. Dante Chair [or Professorship] in Medieval Imagery. Even on first reference, in all but the most formal settings, these may be rendered the Dante Professor of Medieval Imagery and the Dante Chair in Medieval Imagery.