Webster's Dictionary 
(n.) A religious house presided over by a prior or prioress; - sometimes an offshoot of, an subordinate to, an abbey, and called also cell, and obedience. See Cell, 2.
Cyclopedia of Biblical, Theological and Ecclesiastical Literature 
is a religious house occupied by a society of monks or nuns, the chief of whom is termed a prior (q.v.) or prioress; and of these there are two sorts: first, where the prior is chosen by the convent, and governs as independently as any abbot in his abbey; such were the cathedral priors, and most of those of the Augustine order. Secondly, where the priory is a cell subordinate to some great abbey, and the prior is placed or displaced at the will of the abbot. There was a considerable difference in the regulation of these cells in the mediaeval times; for some were altogether subject to their respective abbots, who sent what officers and monks they pleased, and took their revenues into the common stock of the abbeys; while others consisted of a stated number of monks, under a prior sent to them from the superior abbey; and those priories paid a pension yearly, as an acknowledgment of their subjection, but acted in other matters as independent bodies, and had the rest of the revenues for their own use. The priories or cells were always of the same order as the abbeys on which they depended, though sometimes their inmates were of a different sex; it being usual, after the Norman Conquest, for the great abbeys to build nunneries on some of their manors, which should be subject to their visitation.
Alien priories were cells, or small religious houses, in one country dependent on large foreign monasteries. When manors or tithes were given to distant religious houses, the monks, either to increase the authority of their own order, or perhaps rather to have faithful stewards of their revenues, built convenient houses for the reception of small fraternities of their body, who were deputed to reside at and govern those cells. — Hook, s.v. In the fourth year of Henry V, during the war with France, all the alien priories (that is, those cells of the religious houses in England which belonged to foreign monasteries) which were not conventual were dissolved by act of Parliament and granted to the crown. About the year 1540 the cathedrals founded for priories were turned into deaneries and prebends.