Convex and concave lenses work through understanding how light interacts with different mediums, specifically glass or glass compounds, and manipulating the medium's shape to achieve some desired magnification effect. Lenses are designed incorporating convex and/or concave designs, depending on the desired effect and application.
Light travels through a vacuum at 3 times 10 to the 8th power meters per second. When light encounters a different medium, its velocity is affected by interacting with atoms. The amount to which its velocity is affected depends on the properties of the medium, including transmittance, and is called its index of refraction. If not enough light is transmitted, and is instead absorbed or reflected, it will not be possible to see through a lens. That is why a clear, colorless glass makes the best optical lens for vision correction, while darker, more reflective glass is used for eye protection.
A medium's index of refraction (n) is the quotient of the speed of light in a vacuum (c) divided by the speed of light in the medium (v). Using a theory called Snell's law, one can determine exactly how a medium will effect light and shape it appropriately. Convex shapes are used to concentrate light and increase the apparent size of things, while concave shapes are used to disperse light and decrease the apparent size of things.