They are all essentially the same phenomena with different amplitudes... in a sense.
A pressure wave is just a general term for a pressure disturbance moving through a medium. This happens to be exactly what a sound wave is. These disturbances move at the speed of sound in the medium in which they are traveling. There is no formal distinction between the two, as any amplitude of pressure wave could be heard as sound provided the listening device was sensitive enough.
A blast wave would be a pressure wave created by an explosive blast. Depending on the nature of the explosion, it may or may not be a shock wave.
A shock wave is a specific type of pressure disturbance moving through a fluid (or stationary in the fluid depending on the situation. Consider a pressure wave (sound wave) moving through a fluid. For small amplitudes, the wave passes by and the medium more or less returns to its initial state. However, if the amplitude is large enough, it will drag a little bit of the air along with it. That means any sound waves propagating behind it will tend to catch up with the original wave and drag the fluid behind them still faster. That process stacks up and eventually you can have a number of pressure waves coalesce into a shock wave.
Shocks waves are near discontinuous changes in pressure, velocity, temperature and density in the fluid (changes occur over 6 or so mean free paths between molecules). To the stationary observer, they can appear to be moving faster than the speed of sound, though they will
always be traveling below the speed of sound in the medium out of which they are propagating, even if they are traveling faster the speed of sound in the medium into which they are propagating.
When an explosion occurs such that a shock is generated, a few things happen when the shock passes over you. First, there is a sharp rise in pressure, temperature and density, so that pressure spike itself can do a lot of damage. However, at least as important is the flow behind the shock. When the shock passes, it is dragging an awful lot of fluid with it. In fact, the flow being dragged behind the shock can even be supersonic if the shock is strong enough! The potentially high velocities combined with high density behind the shock means that a lot of the real damage correlates with the incredibly high dynamic pressure behind a passing shock wave.
As a quick example, if you were to explode a bomb at ground level such that the shock was moving at Mach 1.8, the simple shock wave that passes by would cause the pressure to spike to about 3.6 times the ambient pressure. Additionally, say the ambient pressure was 0.3 atm and the ambient temperature was 220 K. You would have over 0.5 atm of dynamic pressure behind that passing shock as well due to an induced velocity of just shy of 310 m/s. That is where a lot of the damage comes from.
However, consider a second case. A bomb is usually detonated in the air such that the shock is coming down at the ground at an angle. That results in not only the original shock, but a reflected shock, which effectively compresses the flow twice. The pressure would increase almost 10 times over ambient! The induced velocity doesn't change much, but because of the increased pressure and density, the dynamic pressure would now be roughly twice as high as the single shock case. That is why bombs are usually detonated in the air and not at ground level.
That probably went a lot farther in-depth than you wanted. Oops.
