There are two things that can be checked to find out. The first is the relay that runs the blower motor may need replaced and the second is the motor itself needs replaced.
Depending on the way the vehicle is wired, it could be a bad ignition switch or a stuck relay. You'll need to look at a wiring diagram to find out how power is supplied to the blower and AC system. Then go backwards to find out what's on that SHOULDN't be.
First guess would be a bad ignition switch.
Climate control blower motor was the fix. The part cost 441 dollars.
RH under hood, between blower and evaporator according to diagram. The blower control module is most likely the culprit. It was on my 87 covette ,easy to replace but cost about $150.00
replace the ignition switch
its your squirl cage inside of your blower motor. guaranteed. happened to me twice
the heater matrix is blocked
Open hood . Disconnect blower module located near blower fan. Disconnect, if blower shuts off, repalce module
It usually shouldn't, but if it does, it should only stay like that for a few moments.
One reason a 2004 Monte Carlo heater motor blower may not work is low coolant. You can check the coolant level after the car has been turned off.
Check the fuse.
The blower motor relay switch in the Honda Civic is located at the center of the engine closest to the firewall. The relay is engaged when the ignition is turned to the on position.