The hats worn by rich women during the Middle Ages were very artistic and complex. If you want to know more about the many types I suggest you visit http://www.kats-hats.co.uk/ . Here you can order custom made medieval hats, and read about them. A lot of the pictures in this entry are from that website.
In an earlier entry I wrote about the hair do and showed a couple of pictures of hats, this will be about hats only.
The Hennin was a tall construction and could be formed like a cone, two horns or just a tall hat. Often a white veil was attached to it and I reckon it belongs primarily to the 15th century. When wearing a hennin the hair would rarely show.
To show you some of the reconstruction of these I have a few examples from http://www.kats-hats.co.uk/
There also were a heart shaped and a two horned hennin:
And here reconstructed:
There were a lot of other types but the two others that I would like to show you here are the Toque (12th-13th century) and the Crispinette (1300-1500) The Toque is a bit less pompous than the Hennin and looks more like a normal hat with or without a veil.
The crispinette (or caul) is often worn by movie medieval princesses such as Princess Isabelle in "Brave Heart" or princess Amidala in "Star Wars" (okay not medieval movie, but still). It consisted of golden nets around two buns of hair on each side of the head.