A string dampener is a small rubber mechanism, often circular, that is placed inside of a racket's string bed, near the shoulders and throat of a racket.
The lone purpose of a vibration dampener is to reduce the vibrations off of the racket string bed when it connects with a ball. When a ball strikes a tennis racket, an audible "ping" can be heard from the vibrations. Players that enjoy using string dampeners often use it to stop this sound. For most players, this is more of a way to avoid a mental nuisance than something that affects them physically.
It is commonly believed among novices, and even some experienced players, that vibration dampeners can help reduce the stress on the elbow of a player or that they can reduce the power level on a racket they are equipped on. Many manufacturers of dampeners use these claims in their marketing ploys; however, there is very little evidence that supports either of these claims. In fact, many tennis racket manufacturers build vibration absorption into the frames of individual rackets. This kind of vibration absorption is much better compared to that of the small and insignificant vibration dampeners'.