Simple Definition of Curvature
Before we start formally defining curvature, we must at least have a basic idea of what it should be. So curvature is how curvy something is. What I mean is, if something doesn't curve at all, like y = x, the curvature is 0. If something is extremely curvy, then it's curvature is really high. Now, I know that's very hand-wavy, so let's get to some real math.
So, we need to define a few things.