Compressibility is an aerodynamics term referring to a host of effects that only become evident as an aircraft approaches the speed of sound. These effects, often several of them at a time, made it very difficult for World War II era aircraft to reach speeds much beyond 500mph.
Some of the minor effects include changes to the airflow that lead to problems in control. For instance, the P-38 Lightning had a particular problem in high speed dives that led to the horizontal stabilizer losing "authority". Pilots would enter dives, and then find that they could no longer control the plane which continued to nose over until it crashed. Adding a "belly flap" to upset the airflow cured the problem.
A similar problem effected some models of the Supermarine Spitfire. At high speeds the ailerons could apply more torque than the Spitfire's thin wings could handle, and the entire wing would twist in the opposite direction. This meant that the plane would roll in the direction opposite to what the pilot expected, and led to a number of accidents. This wasn't noticed until later model Spitfires like the Mk.IX started to appear, because earlier models weren't fast enough. This was solved by adding considerable strength to the wings, and was wholely cured when the Mk.XIV was introduced.
The Messerschmitt Bf 109 and Mitsubishi Zero had the exact opposite problem, the controls were too weak. At higher speeds the pilot simply couldn't move the controls because there was too much airflow over the control surfaces. The planes would become difficult to manoeuvre, and at high enough speeds even less manoeuvrable aircraft could out-turn them.
Finally, another common problem that fits into this category is flutter. At some speeds the airflow over the control surfaces will become turbulent, and the controls will start to flutter. If the speed of the fluttering is close to a harmonic of the control's movement, the resonance could break the control off completely. This was a serious problem on the Zero. When they first encountered problems with the poor control at high speed they addressed it with a new style of control surface with more power. However this introduced a new resonant mode, and a number of planes disappeared before this was discovered.
All of the items above are often talked about when the term "compressibility" is used, but in a manner of speaking, they are all incorrectly used. From a strictly aerodynamic point of view, the term should refer only to those effects arising as a side effect of the changes in airflow from a compressible to incompressible fluid as you approach the speed of sound. There are two effects in particular, wave drag and critical mach.
Wave drag is a sudden rise in drag on the aircraft, caused by air building up in front of it. At lower speeds this air has time to "get out of the way", guided by the air in front of it that is in contact with the aircraft. But at the speed of sound this can no longer happen. Air which was previously following the streamline around the aircraft now hits it directly. The amount of power needed to overcome this effect is considerable.
At the speed of sound the way that lift is generated changes dramatically, from being dominated by Bernoulli's principle to forces generated by shock waves. Since the air on the top of the wing is travelling faster than on the bottom, due to Bernoulii effect, at speeds close to the speed of sound the air on the top of the wing will be accelerated to supersonic. When this happens the distribution of lift changes dramatically, typically causing a powerful nose-down trim. Since the aircraft normally approached these speeds only in a dive, pilots would report the aircraft attempting to nose over into the ground.
All of these effects have negative effects on the control or performance of the plane. For this reason it's common to see references to aircraft that suffer from compressibility. The P-38 and Zero are particularly common examples, although in fact they are both bad ones.
