A running martingale is made up of a Y-shaped strap attached to the cinch/girth that runs up through the front legs, passes through a strap encircling the horse's neck and forks into two straps, each with a ring attached for the reins to pass through, thus working off the horse's bit, rather than his noseband.
A running martingale is often used for training young horses in a diverse range of disciplines. A properly adjusted running martingale should allow for a straight line from your hands to the bit. Used in conjunction with a snaffle (it is not recommended for use with a curb bit, harsh) a running martingale discourages a horse from raising his head too high and jutting his nose out to evade the bit's action. Each rein runs through a ring, thus providing a point of leverage to aid in teaching the horse to yield to pressure from the rider's hands. If the horse raises his head above a certain point, the martingale restricts the movement of the reins, which in turn causes the bit to bear down on the bars of the horse's mouth. Rubber "stops" on the reins prevents the rings from sliding too far toward the bit, where they might get caught on fastenings. Another stop is placed where the martingale strap runs from the girth through the neck strap so that the martingale doesn't form a dangerous loop that the horse might put a foot through.
The running martingale is meant to help rebalance a horse by coming into play only when his head gets out of position. It's not meant to force the horse into a restrictive head carriage and is not a substitute for gently hands or effective leg and seat cues.
A standing martingale doesn't exert its influence through the bit but rather from a noseband. it attaches to a cavesson noseband, and a single martingale strap runs between the horse's front legs to the girth. In the English version, the martingale strap passes through a neck strap fitted with a rubber stop to prevent it from sagging between the horse's legs. A standing martingale is known in western circles as a tie-down. A loose-fitting noseband is added under the bridle. The tie-down strap attaches to a ring on the bottom of the noseband at one end and to the cinch at the other. it is common to run the tie-down strap through the ring on the breast collar to secure it. Tie-downs are often used on horses where the fastest time wins. If a horse gets his head too high, it's hard for them to maneuver at speed. A tie-down can help produce balance for a horse during hard stops and fast turns.
A Tie-Down or standing martingale can also help prevent a horse from pushing his nose out to evade the bit and losing his drive from behind. Since it is attached to a noseband, it can accommodate a wider range of bits.
Standing martingales are also found in English disciplines to prevent a horse from tossing his head and avoiding the bit, coming forward out of frame or pulling on the rider. A standing martingale for jumper or hunter accommodates a higher headset than one used on a western horse by coming into play from the horse's chest rather than the girth. Although it attaches to the girth, the English version exerts pressure mainly from the juncture where the neck strap meets the martingale strap. The neck strap should lay approximately where the neck and chest intersect. "These are just guidelines" "Fit can very from horse to horse.
A standing martingale should not be so tight that it forces the horse's head down into an unnatural position. One recommendation is for the horse to be able to put his head forward at least six inches without the martingale coming into play. Also, you should be able to push the martingale strap back to touch the horse's throat. This article written here with permission from Horse Illustrated, May 2003.