Blower motor runs constantly.
I leave my blower running all the time on purpose, to circulate & filter the air and to help even out the warmer/cooler rooms and upper/lower floors, so this isn't necessarily a bad thing.
First thing to check is obviously the simple thing, and that's to see if your fan switch (not the heat/air/off switch) is turned to "on" vs "auto" (I have too assume you already checked that but sometimes the simple answer is the solution). If not, maybe there's a problem with the thermostat. The thermostat, or the fan switch in the thermostat, could be bad. Or possibly some wires are crossing/touching behind the thermostat.
(Disclaimer: I'm not an HVAC tech, I just had this problem before at home and it was a bad thermostat)