Contact lenses can be made of several materials, but the main theme is that it has to be an inert material that is permeable to atmospheric gases. The lens has to allow the eye to "breathe". But in the end it's just a very small lens, shaped like a lens, but adapted to the shape of the cornea, to which it adheres by a combination of surface forces and interaction with the water in the eye.
As for the "amazement factor" regarding its thinness, it has little to do with how thin it is, but how big are its optical properties, like the refractive index. The ability to bend light comes from these parameters.
It's also of note that the smaller the radius of the lens, the thinner it can be overall. Regular lenses can be very thin at the middle and only get thick at the borders. But if you cut a small circle around the middle, it can be very thin overall indeed.