Draw reins and running reins are pieces of riding equipment used for training that use the Mechanical advantage of a 'single movable pulley' to cause the horse to bring its head down and inward. While a regular rein is the strap that attaches to the bit and is held by the rider, these types of reins slide through the bit ring, adding leverage to the rider's hands and arms, allowing the rider to force the horse's head into a desired position.
Usage of the term in English riding and Western riding disciplines refers to slightly different designs that nonetheless work on essentially the same leverage principles.
The terms "draw reins" and "running reins" are often used interchangeably in the English disciplines. The terms most often refer to reins which go from the rider's hand, through one bit ring (inside to outside), and attach to the girth. Usually this style is called a "running rein," though the term "draw rein" is also sometimes used. Some styles attach directly to a regular snaffle rein, which limits the degree of force applied, while others slide freely and with little limit to the leverage that can be applied. Another style, also called a "draw rein," runs from the rider's hands, through the bit ring (outside to inside), over the poll, through the other bit ring, and back to the rider's hands, without attaching to the girth.
In Hunt seat style English riding, these devices originally developed as a two-rein bitting system. One set of reins is an ordinary direct snaffle rein, and the other is the running or draw rein. The rider holds these reins in a manner similar to a double bridle, usually with the snaffle rein below the fourth finger and the running or draw rein between the third and fourth fingers, although there are variations on this. When riding in this fashion, the rider should ride predominantly on the snaffle rein. However, riders and trainers may often be observed using only the running or draw rein, in many cases with the snaffle rein completely absent from the bridle. Riding with draw reins alone is a controversial practice among English riders.
Draw reins in the western riding disciplines are always attached to the rings of the cinch (a western-style girth), usually on each side of a western saddle, run through the bit rings (either inside to outside or vice versa, there is no firm rule, though the rein moves more smoothly if the inside goes to the girth and the outside to the hand), and then to the hands of the rider. They are rarely used in a two-rein system, usually are used alone or used with the regular bridle reins allowed to lay slack and not held by the rider. While use of the standard draw reins presents only mild controversy in western circles, a controversial use of this rein in western riding is its use on a curb bit, a practice that applies incorrect leverage, is dangerous, and creates an effect that some consider abusive.