The three mirrors are arranged to form the corner of a cuboid, or more specifically a cube with the reflecting surfaces facing inwards. An incident ray of light can be considered to have an x, y and z component, where x, y and z are Cartesian components of the ray vector. The ray will strike one mirror first and then be reflected on to another, which in turn reflects on to the third mirror. With each reflection one component of the original ray is reversed. So (x,y,z)→(-x,y,z)→(-x,-y,z)→(-x,-y,-z). In the last reflection all three components of the ray have been reversed, meaning that the ray is reflected back parallel to the path it arrived.
A retroreflector on the moon, for example, would bounce a laser ray back the way it came and by timing the journey from source back to source using accurate timing devices, the distance of the moon to the earth can be calculated because the speed of light in a vacuum is known. Using satellites instead of terrestrial sources gives better results because there is no atmosphere to cause variation in the speed of light.
Retroreflectors on the moon are usually in arrays so that they capture and reflect light over a wide area.
Retroreflectors can be used similarly for navigation systems to provide great accuracy in determining location. So they can be installed in satellites for just such a purpose.
They can also be used to determine the speed of light in various media where distances between points are known and the timing is measured.
They're used in reflectors on the road surface and in vehicle reflectors. They can also be used in hi-viz clothing.
