Cloister and convent are general terms denoting a place of retirement from the world for members of a religious community; they may apply to houses for recluses of either sex.
In such use cloister stresses the idea of seclusion from the world; convent, of community of living.
Basically a monastery is a cloister for monks; in actual use it is often applied to a convent for men or occasionally for women who combine the cloistered life with teaching, preaching, or other work.
Nunnery, which specifically denotes a cloister for nuns, is often displaced by convent with the same specific meaning. A monastery or nunnery governed by an abbot or an abbess is called an abbey; by a prior or prioress, a priory.
A priory is subordinate in rank to, but often independent of, an abbey.