How does exceeding the critical Mach number of an aircraft affect its operation?
At the critical mach number, some part of the aircraft (usually the wing) will have air flowing over it at a speed in excess of mach 1.

If the aircraft is not meant to fly at transonic or supersonic speeds, shock waves will flow over the wing. This can either cause the wing to stall, the control surfaces to become unresponsive, or the plane to go into the mach tuck, which is very dangerous.

The mach tuck is a phenomenon first observed with the P-38 Lightnings and Spitfires in World War II. These planes were so clean that when dive bombing it was possible to exceed their critical mach number. When this happened the center of lift would move back on the wing. This would cause the plane to 'tuck'; the elevators stop functioning, and the plane's dive becomes steeper and steeper until the aircraft is actually partially inverted. Recovery can be attempted by lowering the gear, extending the dive brakes, and lowering the flaps.

In later versions of the P-38 a small 'speed bump' flap was added for diving which caused a more even pressure distribution on the wing, and decreased the chance of entering a mach tuck.