There are three main types of martingale;
i) a standing martingale which uses pressure on the nose of the horse to encourage the horse to lower their head when other methods have failed, it can also be used to protect the rider from a horse which throws it's head up and back. It is not as common as it once was and is now usually confined to polo and hunting as this martingale can't be used while jumping and has to be used with a cavesson or flash noseband. It is also independent of the rider's control while the horse is in work.
ii) a running martingale is the most common type of martingale which offers a rider extra control by preventing the horse from raising its head above the point where it can be controlled by keeping the pressure on the bars of the mouth, it also allows the rider to give the horse full freedom of its head if needs be. Popular for jumping and cross country especially on a strong, young and inexperienced horse, it can also be useful for inexperienced riders who do not have full control of their hand position to prevent them upsetting a horse.
iii) a bib martingale is a running martingale with the straps filled in forming a 'bib'. By having the straps connected this keeps the straps closer together and prevents a horse from grabbing the straps.
Martingales are not permitted for use in dressage competitions of in the dressage phase of eventing.
Breastplates and breast girths are useful in preventing the saddle sliding back while a horse is moving, there are two main types of breastplates 3-point and 5-point. A 3-point breastplate connects at the girth under the chest and to the d-rings of the saddle while a 5-point breastplate also connects at the girth straps, both of these allow a running martingale attachment to be used and are very popular with eventers.
Popular in horse racing and increasingly popular with eventers and show jumpers; a breast girth is an elasticated strap which connects from girth strap to girth strap around the front of the horse with a neck strap to hold it in place, a breast girth does not connect under the chest and it has no way to attach a martingale.
It is not uncommon for a breastplate and breast girth to be used in combination by eventers during cross country.