Five-year-old split system consisting of a gas furnace to provide heat in the winter and a condensing unit and "a" coil to provide cooling is blowing warm air. The thermostat is set in the coolng mode with the set-point below the room temperature, and the fan switch is set in the AUTO position. The condenser fan is running normal but the compressor is not running. After shutting off the unit the compressor is hot to the touch. What can be the problem?

I think your compressor in the outside unit is toast. The only way to test this is with a amprobe. It is also possible that the unit has lost all or part of its refrigerant charge allowing the compressor to overheat. Time to call a professional to get a better feeling as to what your next step will be.