Plain English Meaning
When an aircraft flies fast enough that the air ahead of it starts to bunch up rather than flow smoothly around it, the aircraft starts behaving differently — drag rises sharply, lift shifts, and the controls feel different. These are compressibility effects.
Full Definition
Aerodynamic changes that occur as an aircraft approaches the speed of sound, where air can no longer be treated as an incompressible fluid. As airflow over parts of the airframe nears Mach 1, the air begins to compress significantly, producing shock waves, sudden increases in drag, changes in lift distribution, and altered control response. These effects set a practical upper speed limit for high-altitude, and are a factor in at altitude.