Nosebands: "Your horse still has to be able to eat a treat."

From: http://www.eurodressage.com/2012/02/08/noseband-special-part-ii-purpose-noseband

"In the training of a recently broken horse the noseband plays the important part of showing the horse how to handle the pressure the right way when the bit action sets in. With a young horse it is a natural reaction that it opens the mouth or crosses the jaw too much when it feels bit pressure on the tongue or bars for the first times. The youngster tries to escape the first (soft) contact and braces the chewing musculature which should relax. By closing the mouth in an acceptable manner the young horse will more easily learn to take the bit and to chew which is only possible if the noseband is not too tight. When riding in a simple broken snaffle bit the rein aids are transported to the bars and the tongue. The horse which tries to evade the bit gets some pressure on the nose by the noseband and gives in, ideally by lowering the head. This means that the noseband, according to the type, takes over some pressure from the bit. The general rule is: the lower a noseband is positioned the more pressure is taken over. While the nose strap gives only some pressure on the sensitive nose of the horse, the chin strap supports the lower jaw.

A correctly fitted noseband not only helps to show the youngster the right acceptance of the bit, but also prevents that it establishes unpleasant reactions like gaping, crossing the jaws or even putting the tongue over the bit which can happen no matter how good the rider’s hands are.

Renowned classical French dressage rider Catherine Henriquet gives an easy general tip to find the correct tightness of a noseband: "your horse still has to be able to eat a treat." Under the tutelage of her husband Michel, a long time student of legendary Nuno Oliveira, Catherine became an Olympic dressage rider strictly following the classical principles."

Nosebands for the driving horse

Although the advanced, well trained horse could be driven in quiet, well controlled environments without a noseband, or even without a bridle altogether, in practice, a properly fitted noseband serves mainly as a safety mechanism for achieving control with the bit in the chance that the horse reacts badly to circumstances beyond the control of the driver.  Much like the kicking strap, it is adjusted so that the horse barely is aware of it's presence until it is needed to regain control when the horse is frightened. ~ Jeff Morse

More:
Fascial connections of peripheral nerve of the horse
On the Ignorance of Noseband Tightness and Vague FEI Noseband Rules
Noseband Special: Part I: The History of the Noseband
Noseband Special: Part II: The Purpose of the Noseband
Noseband Special: Part III: Riders and Trainers on Their Choice in Noseband
Noseband Special: Part IV: The Thicker, the Wider, the Better?
An Investigation into Noseband Tightness Levels on Competition Horses
ISES Suggest to Empower FEI Stewards to Control Tightness of Noseband
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