An abbey is a church that's part of a monastery or convent. You'll often find monks or nuns in an abbey.
There are a few types of abbeys, but they're all related to intense religious dedication. An abbey can be a monastery or convent, which is either composed of monks or nuns who have withdrawn from society to live a life of poverty and chastity. You can also call a church associated with a monastery or convent an abbey. That type of abbey will be run by the monks or nuns but also open to other churchgoers.