Abbey and Monastery are often seen as common terms, but both terms are not totally the same. Monastery refers to a building where monks, nuns, clerics, or others who have taken certain religious vows live together. The way a monastery is designed is that as it grows, it becomes an abbey.
Basically, a monastery is a place designed for those who have prepared to live a monastic life. An abbey refers to a society of people or a bigger community of people who have devoted themselves to certain religious lifestyles, which is headed by an abbess or abbot. If an abbey is occupied mostly by monks, the superior or head of such abbey is usually referred to as an abbot.
On the other hand, if an abbey is occupied mostly by nuns, the head of such abbey is usually referred to as an abbess. In the event that you have both abbots and abbesses as leaders in an abbey, both usually have certain territorial responsibilities that are well defined.