A martingale is a piece of tack commonly used in everyday riding as well as competition.
It is primarily used to help prevent the horse from carrying or throwing his head too high and the running martingale acts as a gentle guide, the standing martingale being a more assertive relation. The neck strap is useful as a safety grasp and wearing the martingale just offers a little more control in case of misbehaviour.
The correctly fitting running martingale should allow plenty of movement before coming into action.
Breastplates with running attachments offer the same benefits as a running martingale, but attach to the saddle, at several points (depending on the style of breastplate) offering stability and helping prevent the saddle from slipping during competition when things can become a little awry as things do when moving at speed and unpredictability rules the day. They are a good piece of safety equipment.