Refraction occurs when the energy of an incoming light wave matches the natural vibration frequency of the electrons in a material. The light wave penetrates deeply into the material, and causes small vibrations in the electrons. The electrons pass these vibrations on to the atoms in the material, and they send out light waves of the same frequency as the incoming wave. But, this all takes time. The part of the wave inside the material slows down, while the part of the wave outside the object maintains its original frequency. This has the effect of bending the portion of the wave inside the object toward what is called the normal line, an imaginary straight line that runs perpendicular to the surface of the object. The deviation from the normal line of the light inside the object will be less than the deviation of the light before it entered the object.
The amount of bending, or angle of refraction, of the light wave depends on how much the material slows down the light. Diamonds would not be so glittery if they did not slow down incoming light much more than, say, water does. Diamonds have a higher index of refraction than water, which is to say that they slow down light to a greater degree.
One interesting note about refraction is that light of different frequencies, or energies, will bend at slightly different angles. Let's compare violet light and red light when they enter a glass prism. Because violet light has more energy, it takes longer to interact with the glass. As such, it is slowed down to a greater extent than a wave of red light, and will be bent to a greater degree. This accounts for the order of the colors that we see in a rainbow. It is alos what gives a diamond the rainbow fringes that make it so pleasing to the eye.
Definition of Index of Refraction taken from http://science.howstuffworks.com/light12.htm
For a more technical description you chek out the explanation at http://en.wikipedia.org/wiki/Refractive_index
Now why is this importantí When creating materials, you need to take Index of Refraction into account. Otherwise, your materials will not be convincing. And with the new transmissivity feature added in the last release in addition to the IOR settings panel already in place, blender is capable of convincing IOR values.
Here is a list of IOR values for your reference. This image is rather small, for a clearer copy please chek the pdf included with this issue.