You should already be familiar with damping. It simply refers to the fact that if you set a spring going, it eventually stops.
The wikipedia article should cover most of what you want to know.
Any particular spring may be damped for all sorts of reasons. Any way the spring can lose energy contributes to damping, so it could be lost internally to heat due to stressing the material, or externally to heat via friction, or externally to an electromagnetic field, to some sort of mechanical dashpot, etc.
If you were designing an experiment to study damping, you could be interested in a number of different particular things. You will have to make your own choice about what the most interesting thing to study is. For example, would you like measure the damping ratio? The overall magnitude of the damping effect? The time it takes your spring to reduce its amplitude to 1/2 its previous value?
If you have a particular effect picked out, do you want to know how it depends on the load on the spring, or the material of the spring, or whether the spring is in a vacuum, etc. There are many possible things to study, so your question has no definite answer. I'm sure you can find something particular you find worthwhile.