Why is my indoor fan running but my heat pump is off?

Heat Pumps are designed to extract every bit of available heat in order to attain maximum efficiency. When a heat pump shuts off, the indoor unit still has a residual amount of heat stored in its coil – by blowing the air over this, all stored heat is taken out of the coil.
This energy saving feature is enabled when the fan is set to “Auto” mode. In addition, the fan can be set to run continuously in the “On” mode creating perpetual airflow through the ductwork, and to better mix air in the home providing more consistent temperatures.