It's called a bearing rein but it causes severe pain to the horse and I strongly recommend you don't use it.
Holding a horses head up high is considered a bad thing to do as it interferes with the horses ability to balance, breath and move. That being said the Bearing rein has been used for hundreds of years to get the 'high headed' look on carriage horses. This piece of equipment is cruel and should not be used on horses.
The piece is called a bearing or bering rein(s). This used to be a standard piece of tack before the automoble. The horse was forced to hold his head at an unaturally high level putting strain on the entire topline* of the animal. It was more of a fashion statement, people thinking the horses looked better while pulling the family's carriage. *The topline of the horse is the poll, crest, withers, back, croupe and dock.
if you are referring to horses, it is not called a "cheek strap". It is called a "cheek piece". A cheek piece is part of the bridle. When on a horse, it is along the side of a horses face, basically where the horses cheek is (from the ears to the muzzle). It is joined to the Bit (in the horses mouth) and the head piece which is the part of the bridle that is at the top of the horses head, by the poll (behind its ears). I hope this was helpful information.
If you are riding a horse and you want to force its head higher, that would be done with the reins. Holding the reins tightly or high up on the horses neck is considered poor horsemanship and can get you kicked out of a competition quickly. Each horse carries it's neck and head according to it's build,this can be modified through training, or you could just buy a horse with a higher neck/head set. Remember though that not allowing a horse to use it's head and forcing it to stay up can cause a multitude of health and attitude problems later on.
running martingale
The Irish Martingale.
A tom thumb is use on some horses who hold their head higher than most people like them and their for horses that though their head like idiots.
A headstall for a horse is a hanger put on a horse's head used for holding a bit,bosal, nose piece, or other piece of equipment to control a horse while riding. When a headstall is combined with a bit (or bosal, nose piece, etc.) the combination is known as a bridle. o OK now you know
Most of the time it's the horses head.
A horse can throw it's head back no matter what piece of tack is used if it's determined to do it. But many people use a martingale to keep a horses head down, though this is not a good substitute for real training of the horse.
The shank in real life and on howrseAn Irish martingale.
Although the part of a horses head have special names, the head itself does not - It's just a head. Mfire