Dynamic range, in the context of audio, is the difference between the loudest sound and the quietest sound. It is often described as a ratio, and commonly notated in decibels (dB). Think of it this way: You are watching a movie where the character whispers, but it's just as loud as the big explosion at the end. This would be an example of low dynamic range.
Dynamic range is very easily noticeable when listening to music, and in modern music compressors are used to limit dynamic range, making everything seem very loud. This causes ear fatigue, and means that listening for long periods of time is not as comfortable.