A natural example of this is as a phase in a sound wave or phonon. Half of a sound wave is made up of the compression of the medium, and the other half is the decompression or rarefaction of the medium.
Another natural example of rarefaction is in the layers of our atmosphere. Because what constitutes our atmosphere has mass, it is definite that most of the atmospheric matter will be nearer to the Earth. Therefore, air at higher layers of the atmosphere is considered to have less pressure, or is rarefied in relation to air at lower layers.
Rarefaction can be easily observed by compressing a spring and releasing it.
Rarefaction waves expand with time; for most gases the rarefaction wave keeps the same overall profile at all times (it is a 'self-similar expansion'). Each part of the wave travels at the local speed of sound, in the local medium. This expansion behaviour is in contrast to the behaviour of pressure increases, which get narrower with time, until they steepen into shock waves.
Rarefaction can refer to an area of low relative pressure following a shockwave.