A stavropegial' (also spelled stavropigial or stavropighial) institution, usually a monastery, is one which falls directly under the omophorion of the primate of a church rather than under the local diocesan bishop.
The Greek term σταυροπηγιον literally means "fixture of a cross," and referred to crosses used to mark boundary points. In the liturgical context, a stavropegion was a cross fixed by the bishop on the side of a new church. The term came to be employed mainly to refer to monasteries which owed canonical allegiance to the Patriarch of Constantinople.
In a number of Byzantine documents from the 10th to 14th centuries, the term stavropegial was used roughly synonymously with patriarchal when referring to monastic communities of this sort, though in some cases, the two terms were distinguished, with stavropegial referring only to those monasteries which had actually been founded by the patriarch.
Stavropegial monasteries in the Byzantine period acknowledged the authority of the patriarch as their bishop, commemorated him in liturgical services, and paid him the kanonikon, an ecclesiastical tax which provided a significant source of income for the patriarchate.
- Oxford Dictionary of Byzantium, pp. 1946-47