I think it is just how thick and stiff the felt is made (in felt hats). Some hats have wire in the brim, but as far as I know only in the very edge so the brim keeps a certain shape. If the hat is made from cloth it may use cardboard or very stiff fabric sewn between the softer fabric to give it stiffness.
I forgot the titles, but I once got a couple of books from the library which all explained making various hats from scratch. So I would suggest checking your library. How you make a hat depends a lot on what kind of hat you want to make. There are a lot of different ways to make one. It really depends on material and what final look you want what you could try. For felt hats you usually need hat blocks and there is a different block for the crown and the brim.
