The ignition is turned off, yet the engine continues to run.  What is happening??

 

You are experiencing a phenomenon known in the industry as afterrunning or dieseling.  Here is a definition of afterrunning:

 

Afterrunning

Also known as dieseling. When a spark-ignited engine continues to run after the ignition has been turned off. There are two basic causes of afterrunning: surface and compression ignition. In surface ignition, the surfaces of the combustion chamber remain hot enough to provide a source of ignition after the spark ignition is terminated. In compression ignition, the conditions of temperature, pressure, fuel composition, and engine idle speed allow ignition to continue.

 

What it basically means is if the conditions are right and fuel is supplied to an engine it can continue to run with no source of ignition (spark plug spark).  The key here is "fuel is supplied".  Most of the time it is aided by a mis-adjusted carburetor.  Most carburetor idle control is performed by a solenoid which holds the throttle open at idle so that the engine doesn't stall.  It is the adjustment of this solenoid plunger that sets the idle speed.  When the key is turned off, the solenoid is deactivated allowing the throttle to close to shut off the flow of air and fuel into the intake manifold.  It closes to the "bottom stop" a screw which prevents the throttle plate from sticking in the throttle body.  A lot of time someone attempts to adjust the idle speed using the "bottom screw".  When this happens the deactivation of the solenoid has no effect on the throttle position.  The throttle remains partially open allowing fuel to enter the intake manifold and, given the right conditions of engine temperature, keep the engine running.  Hitting the throttle quickly allows a lot of air to rush into the intake manifold causing the mixture to become lean and the engine dies.

 

The fix for this problem is to adjust the idle speed correctly, using the solenoid and not the bottom screw.