I've been asked a few times what I use as the framework of my hats and boxes. It is called 'plastic canvas' and looks like this:
As you can see, it cuts pretty cleanly and it is very flexible, but firm. Because it has those little squares it is best to do an underlayer of a material (usually I use white if I'm layering that with something light, and black if with something dark) so that it is not going to be seen through the top material. This also ensures that there aren't any pointy or ragged edges. You can buy it in sheets or circles (usually I use the circles for mini hat brims and tops) at fabric stores like Joanne Fabrics.
Examples of things made with plastic canvas: