Here is the problem. Lets say it is 80 degrees. I turn the central air conditioner on, set at 78 degrees. The house cools down to 78 degrees. The thermostat appears to shut the blower (from the furnace) off and acts like it everything is shut off. However, the outdoor AC unit continues to run and will continue to run until someone turns the circuit breaker switch off. The unit (and thermostat) is about ten years old and nothing on the system has been replaced recently.

Anyone have any suggestions as to what the problem is?

I'd start by checking the contactor in your outside condensing unit. If possible, next time it happens disconnect one of the small 24 volt lines that goes to the outside condensing unit. If you unhook the wire and the unit still keeps running, your contactor most likely has bad contacts that are sticking closed.

Air conditioner continues to run,but not cooling the house. What could be possible causes?