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.
Contact lenses are small prescription lenses, worn in contact with the eye. They are designed to correct refractive errors and maintain ocular health. They float on the tear film layer on the surface of the cornea. Modern contacts are much more than small eye glass lenses that fit onto your eyes. They do, however, function much like regular eye glasses refracting and focusing light so that objects appear clearly. Since the lenses stick to the tear fluid on your eye surface, they move naturally with you. This is but one advantage contacts have over glasses.