The coefficient of restitution tells you about the energy lost in the collision. Specifically e^2 is the ratio of the kinetic energy after to before the collision in the zero momentum frame. This depends not only on the elastic properties of the material, but also the structure of the body.
If you take a very simple example and have 2 springs hit each other head on then they will compress up to a point then begin to separate. At the point of separation the springs will still be compressed and therefore hold energy kx^2/2 each. This is the energy lost in the collision. If the springs were completely undamped then they would go on oscillating forever never losing this energy. In reality the spring will lose it's energy over a few cycles (as heating of the material) and in fact loses some during the initial compression phase also.
This is essentially what happens in a real collision of 2 objects. Perfect elasticity would imply one so 2 things:
- A body can deform in such a way that it returns all of it's gained energy - this is against the 2nd law of thermodynamics since work is done in deforming the body, or;
- A body can instantly reverse direction with no deformation. This implies an infinite force that acts over zero time
All of the above is constrained to classical mechanics since the quantum reality is somewhat different.
So the coeff of rest tells you about the combination of material and structure. As an experiment you can drop a tennis ball from a height, then cut out a small patch from it and drop that from the same height.