Began in the late 1500's, the convent was originally built to house nuns (and their servants) from the upper-class of Peru. Over the following 4 centuries, the complex eventually became a small city within the center of Arequipa, boasting a maze of rooms, streets, chapels, kitchens and beautiful gardens. However, while a few nuns still live there today, they now occupy only the northern corner. The rest of the convent has been turned into a beautiful public museum. FOR YOU!
The museum is inexpensive (about $10) and laid back. You are free to wander around for as long as you want, but I believe there are guides too. Inside there is also a cafe and gift shop selling religious objects, although I have to admit I haven't really checked these out fully. Oh, and as a bonus: I've even see advertisements for special night-time tours (I can only imagine how breath-taking/romantical it is at night).
It is definitely worth visiting, if only for the incredible photography. The vibrant colors make it really difficult to take a bad photo, and you'll get to feel really good about yourself when you later look at these pictures. Trust me.
But aside from that, the museum has some really cool displays of colonial furniture, art and technology. Plus it is nice to have a quiet area to just enjoy the day and avoid the craziness that is the plaza de armas of Arequipa.