Imagine you use Cycles and you are in the node editor. You use a mix shader and connect two other nodes into its color input, for instance two glossy shader with different colors. Now you take the output of the layer weight node and put it into the fac input of the mix shader.
The result is, that which of the two shaders dominates depends on the angle from which you look onto the object. If the normal of one of the faces of your object is parallel to your viewing direction the upper shader is at 100 % dominance and the wider the angle between this normal and your viewing direction gets, the more the lower shader's influence rises.
Real life occurence:
This looks strange to me in the sense, that I am not used to materials with such optical properties in the real world, I do not even know if I have ever seen such a material knowingly. However, I would like to know if there are typical objects in the real life which behave in such way, so I could try to emulate them in Blender.