And no, the proper term is not "nunnery."
First of all, a convent is based off a Latin word that means “to convene or gather” and can technically refer to a building where a group of priests or religious “come together” after working in various active apostolates. In the United States and many English speaking countries this term is typically used exclusively when referring to the living quarters of an active women’s religious order. However, historically it also referred to men’s religious houses as well.
On the other hand, a monastery is based off a Greek word that means “to live alone” and is normally applied to a building used by a religious order that is “set apart” from the rest of the world. Men religious who live in a monastery are usually named “monks,” and come from a long tradition of monastic living.
Originally monks lived in solitary huts that were close in proximity and only came together for meals and prayer. Over time this developed into a single enclosed building that contains individual rooms for the monks to live in.
Monasteries can contain either men or women religious who are dedicated to a life of prayer that is set apart from the rest of the world. Most of the time ordinary lay people will never see a monk or nun who lives in a monastery as they don’t frequently leave, unless for a special reason.
The word “nunnery” is a slang term for any community of women religious and often refers to either a monastery or a convent. In Shakespeare’s time, the term had a double meaning that adds complexity to Hamlet’s mad ravings to Ophelia: a nunnery could be a convent of nuns, but in profane usage it could also be a brothel.