Compressibility includes the changes in density in the flow. This is important in the flow of air or gases
There are two regimes for considering compressibility:
M < 1
The transition from incompressible flow to compressible flow is at (practical) M = 0.3. Usually if the highest Mach number is less than 0.5, we can ignore the effect of compressibility.
Compressibility affects both lift and drag with effect on the latter being more prominent
M > 1
This is supersonic flow
The effects of compressibility are serious
It includes shocks that cause drastic changes in fluid properties