What is the significance of the speed of sound to aerodynamics and aircraft design?
Your question shows that you know that the speed of sound is the speed of small pressure changes in a fluid, and moving faster than that will give the fluid ahead no warning of an approaching disturbance. I will, therefore, dig a little deeper in my answer, fully knowing that I can only scratch the surface. I encourage you to ask more specifically if one topic piques your interest but is not sufficiently explained.

Short answer

The speed of sound is significant in aircraft design mainly in:


Splitting the flight regimes into two areas with markedly different flow behavior,
helping to convert a geometry for compressible flow to one which can be calculated by simpler, incompressible methods,
limiting the maximum practical flight speed for efficient air travel and
giving the designer hints how to push this boundary up.


In the combustion of gasses the speed of sound is of less significance than the speed at which a flame advances. This is more complex (PDF!) and depends on the turbulence level, mixing ratio, temperature and the activation energy for the combustion reaction to start.

Explanation

To explain the influence of the speed of sound let me look at the molecular level: Air is a collection of molecules which zip around constantly, bumping into each other frequently. We express this with macroscopic parameters:


The intensity of the collisions is measured as pressure,
the number of molecules per volume is measured as density, and
the speed of the molecular movement is measured as temperature.


At rest, the frequency and intensity of these collisions is the same in all directions. If something moves through the gas at a speed far below the speed of sound, the collisions are still roughly of the same intensity all around. The local intensity may vary, but the molecules will stay equally distributed over the volume. This means that the disturbance will cause a change in pressure, but density will be unaffected.

Similarity methods

Moving faster, the disturbance will cause more intense collisions in the direction of movement and will start to push the molecules together: Now it will also cause weak density changes, and those grow with the square of of its speed. To regain an equal distribution over space, the molecules will now move sideways (and up and down), and the pressure changes will reach farther out to the sides. When you compare the pressure fields at different speeds, the "faster" field will look like one made by a more narrow disturbance at lower speed. This effect is captured by the Prandtl-Glauert factor $(1-Mach)^2$. Scaling the Y- and Z-coordinates of an aircraft geometry (where X is in flight direction) with this factor will result in a geometry which can be calculated with incompressible methods.

At the speed of sound this factor has a singularity, so its application should be restricted to speeds below Mach 0.7. Above Mach 1 a similar transformation can be applied, now using the Ackeret factor $(Mach-1)^2$.

Convergent and divergent flow

The ratio of density changes to pressure changes continues to increase such that in supersonic flow the density changes become dominant. This shows when looking at the flow in an expanding or contracting duct:


Subsonic flow in an expanding duct will slow down to preserve density and will accelerate in a contracting duct.
Supersonic flow in an expanding duct will accelerate further, reducing density and pressure, and decelerate in a contracting duct.


This different behavior is used when accelerating the compressed gasses flowing from an aircraft turbine: First the nozzle diameter is reduced until the gas accelerates to the speed of sound. To accelerate the gas further, now the diameter has to increase again in flow direction. The result is a convergent-divergent nozzle, a typical part of supersonic jet engines.

Wave drag

Now picture the aircraft in slow motion from the side as it slices through the air at supersonic speed: Since pressure increases while the aircraft body expands and decreases once it contracts, you will have two drag components in supersonic flow which do not exist in subsonic flow: Forward facing parts experience higher pressure and rearward-facing parts experience suction. This effect is called wave drag and makes supersonic flight inherently less efficient that subsonic flight.

Transsonic design

The same effect also affects the pressure field around an aircraft which is moving close to the speed of sound. Its movement displaces air and has an effect similar to a contracting duct, accelerating the flow in its vicinity as long as the aircraft grows thicker in flow direction, and decelerating it again over its rear half where it contracts. Once this local flow reaches the speed of sound, it will accelerate further while the aircraft contour contracts, leading to a supersonic pocket of air which will not slow down. It does so eventually, collapsing in a local shock which can cause flow separation and high local drag. Even worse, this separation will affect the whole pressure field and reduce the speed increase, which opens a feedback loop which will make the shock fluctuate. This oscillating shock can excite vibrations, render control surfaces ineffective and can incite flutter.

To reduce the local speed increase the surfaces of airplanes can be swept so the flight speed can be increased without running into the aforementioned transsonic effects. Also, sweep will smear out the sonic effects when accelerating through Mach 1, so the drag peak around Mach 1 can be greatly reduced.

Around Mach 1 it is important to limit the rate at which the aircraft's body expands or contracts over the direction of flight, and the wider-ranging nature of pressure changes allows the designer to expand one part of the aircraft to compensate for the local contraction at a different part when both are at the same position relative to the direction of flight. The consequence of this is the area rule which shows in coke-bottle fuselages or big flap track fairings.