Conway was built by Edward the 1st, and it represents one of his great castles, and it is well worth a visit. Besides the castle itself, the city walls encircle a large portion of the town. These walls are open to the public, and nearly complete, and are well worth the additional time required to explore.
I could go on, and plagiarize a ton of things, but for a good, informative web site take a peek at http://www.castlewales.com/conwy.html
Public bathrooms are available.