A royal family is the immediate family of monarchs and sometimes their extended family.
The term imperial family appropriately describes the family of an emperor or empress, and the term papal family describes the family of a pope, while the terms baronial family, comital family, ducal family, archducal family, grand ducal family, or princely family are more appropriate to describe, respectively, the relatives of a reigning baron, count/earl, duke, archduke, grand duke, or prince. From Wikipedia