It's complicated =) At least according to the wikipedia stuff, a Cloister is an enclosed garden (bounded by a covered walkway all around, and a Monastery is the entire complex (and in Catholicism implies a community of men, rather than women).
There's also "Cloistered" which means the monk/nun doesn't interact with the outside world, which can lead to the confusion-- "Cloister" being an architectural element of a Monestary in which a monk could be Cloistered. (I hadn't realized that a Cloister wasn't a nunnery for Cloistered nuns...)