What does "Compressible Flow" mean? Isn't any gas compressible?

It is a question I have asked myself a couple of times, so when I got a good answer to this one, I thought it might be useful to share it. Explanation courtesy of Biswas Souvik, Aspentech.

Compressible flow is definened as any flow where the density is NOT constant along the flow field. If in a flow field where density is completely constant that flow is called incompressible flow. This should not be confused with compressible fluid though. For example, air is a compressible fluid but if there is one flow field where air is flowing but its density is not changing then that flow will be considered incompressible. In reality, few flows are exactly incompressible because density always changes even if by a minute amount. So there is basically a rule of thumb to differentiate between compressible & incompressible flow: if density changes by more than 5% then the flow is compressible, if not the flow is incompressible. For ideal gases with constant Cp/Cv ratio of 1.4, under adiabatic flow conditions, it can be theoritcally shown that for an 5% change of density the Mach No. has to be >= 0.3. So there is another generally accepted rule of thumb,saying if Mach No.>= 0.3 the flow is compressible, otherwise incompressible.