It is most likely burnt contacts on the fan relay, or a bad printed circuit board that is attached to.
OR...the thermostat is set to FAN ON, instead of Fan Auto.
Tell me about how old the system is please. and the full model number on the units ID tag.
Please attach a well focused, close up photo of the wiring diagram glued inside the service panel,you can use the paper clip-icon at the right end of the toolbar at the top of this dialog box,Include a picture of the thermostat wire connections to the printed circuit board in the same area.
If there is no paper clip-icon, look for the '+attach files' link at left of your send button.
I can mark it up with trouble shooting instructions. You will need a multi meter ($30 at a hardware store) and some experience working with appliances, let me know a lot about your skill sets in those regards.
If none of this in your skill sets, tell me which direction you are from the center of the nearest city and I will look for a few well rated service outfits in your area.