The Wikipedia defintion of a shock wave pretty much sums up all I've found online about what a shock wave is:
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium (solid, liquid, gas or plasma) or in some cases in the absence of a material medium, through a field such as an electromagnetic field. Shock waves are characterized by an abrupt, nearly discontinuous change in the characteristics of the medium. Across a shock there is always an extremely rapid rise in pressure, temperature and density of the flow…. A shock wave travels through most media at a higher speed than an ordinary wave.
To me, however, this doesn't seem to provide a very rigorous definition that would allow me to look at a bunch of propagating disturbances and be able to clearly classify it as being a shock wave or (as Wikipedia puts it) a "normal" wave. Although this definition provides a qualitative definition of what sets a shock wave apart from a normal wave, I am wondering if there is a definite difference between a shock wave and normal waves that would allow me to definitively classify a wave as one or the other or if there is a continuous spectrum of wave properties between normal waves and shock waves with no clear boundary between the two (like the electromagnetic spectrum, with only arbitrary boundaries being drawn between the various classes of EM waves).
Best Answer
The first thing that distinguishes a shock wave from an "ordinary" wave is that the initial disturbance in the medium that causes a shock wave is always traveling at a velocity greater than the phase velocity of sound (or light) in the medium. Notice that I said light - that is because there is also a kind of electromagnetic analogue to a shock wave known as Cherenkov radiation (Wikipedia article is here )that is created when a charged particle travels through a medium at a velocity faster than that of the phase velocity of light in the medium (which for many media is some fraction of c).
So getting back to acoustic waves in a gas, the main characteristic that divides a shock wave from an ordinary wave is the thermodynamics of the changes in pressure and temperature due to the wave. For ordinary waves (disturbance less than the phase velocity of sound), the compression and rarefaction of the gas does not entail a change in entropy of the gas - thus an ordinary wave is a reversible process thermodynamically speaking.
For shock waves, this is not the case. The process of compression and rarefaction caused by a shock wave is an irreversible process - it leads to a change in entropy of the gas.
Why is this the case ? Without going too deep into the mathematics, it relates to your question as to how distinct shock waves are from ordinary sound waves. The zone of discontinuity is quite sharp between the disturbance and the shock waves, and the changes in pressure, temperature, and density are large enough that dissipative effects like heat transfer and gas friction come into play.
The boundary conditions involved in analyzing shock waves are known as the Rankine-Hugoniot conditions. The Wikipedia article on Rankine-Hugoniot conditions is actually more detailed about explaining shock waves than the Wikipedia article on shock waves itself.