To amplify those slightly, the performance class relates (related?) to the performance after an engine failure.
Class A requires the aircraft to have adequate performance to account for an engine failure at any stage of flight, including, for example, the most critical point on the take-off roll.
Class C requires similar accountability from about 200 ft in the climb out to about 200 ft on the approach. For the first and last 200 ft, you may have no option but to close the throttles and crash straight ahead.
Class D makes no demands in the case of engine failure.