The compressor on the a/c turns off but the fan continues to run. Tonight when I turned off the a/c at the t-stat, the fan and compressor came back on even though the unit was turned off. What is my problem? The unit is about 24 years old and the t-stat was replaced a few months ago.

When you say "the fan continues to run" do you mean the condenser fan?
Is the thermostat that was replaced the same as the older one? Has this just started to happen, or is this the 1st run of the unit that you've notice it?

The condenser fan runs.
The unit has been doing this for some time, before the new thermostat was put in. It is a different thermostat than the original that was 24 years old.
The turning on after the thermostat is turned off happened for the first time recently.