The blower motor resistor controls the speed of the blower motor. If the blower motor resistor fails, the blower motor will run at full speed all the time.
Blower Motor Relay
The blower motor relay supplies power to the blower motor. If the blower motor relay fails, the blower motor will not receive power and will not turn on.
Wiring
The wiring to the blower motor can also be a cause of the blower motor staying on. If the wiring is damaged or loose, it can cause the blower motor to stay on even when the ignition switch is turned off.