When the ignition is turned on but the engine not yet started, a current flows through the ignition circuit including the ignition light to the alternator to excite the field coils. You then start the car and as soon as the output voltage of the alternator is greater than that of the battery this current stops and the ignition light goes out.
For your horn to sound when the ignition is turned on and then stop after the engine starts means that it is picking up voltage from this initial starter system. It should not be doing that and somewhere in your wiring you have a short.
The very fact that it doesnt sound when the ignition is off suggests that its unlikely to be a problem when the car is parked up. But you do need the services of an auto electrician and likely he will need to go behind the dash.
