What Exactly is a Shock Wave?

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.

Well, shock waves are kind of inappropriately named because they do not oscillate, they are actually a discontinuity. A shock wave is the final stage of a nonlinearly steepening wave that has reached a balance between steepening and energy dissipation (i.e., irreversible energy transformation). An contrasting example would be water waves, where there is insufficient energy dissipation to limit steepening resulting in wave breaking (e.g., white caps or waves that surfers like).

In regular fluids like Earth's atmosphere, energy dissipation arises from binary particle collisions that transform the bulk flow kinetic energy into random kinetic energy (i.e., heat). Sound waves can produce heat as well, but the primary difference is the abrupt and sustained increase in bulk flow speed, density, and thermal pressure across a shock wave (Obviously, if one moves far enough away from a shock, the system will asymptotically return to a quasi-equilibrium with the background... but that's being nit-picky).

In short, shock waves are very different from "normal" waves, using your terminology.

Update
I have a more exhaustive answer here that gives a better explanation of what exactly is a shock wave and how they can form.