It's simply a peice of nylon, rope leather etc..on a halter or headstall that goes across the horses cheek.
They are called check pieces. They are there to hold the bit in your horses mouth.
The purpose of the cheek pieces on a bridle is to hold the bit and browband.
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.
A bridle has 7 parts The reins The Headpiece (Crownpiece) The Browband The Throatlatch The Cheek Piece The Bit The Curb Strap
The noseband of the bridle is the circular strap of leather that goes over the nose, just below the cheek bones. The noseband is used to help guide the horse and hold the bridle in place.
The bridle is a piece of headgear used on horses to help a rider guide the animal. The bridle is the whole piece of equipment. Synonyms for bridle include check, curb, hackamore, halter, leash, rein, and trammels. There are double bridles where the horse has two bits (this is the part that goes into the horses mouth). There are two kinds of straps involved here, the curb strap and the lip strap. The curb strap, also curb chain, runs from one side of the bit to the other and puts pressure on the chin groove when reins are tightened. Lip straps are used on the smaller of the bits. This one prevents the horse from grabbing at the shanks (piece that stabilizes the bit) with its lips.
rein
There is no such thing as a flash bridle, but there is a flash noseband. This noseband is a normal noseband with an extra strap added, which goes around the horse's mouth in front of the bit. This is used on horses who open their mouths a lot, or eat on trails.
A flash bride is just a snaffle type bridle with a little piece of leather on the noseband that holds a flash strap.
a spicific bridle designed for dressage horses in compatition
Saddle strap is a fastener used to attach a saddle on a horse. It is usually made of a leather. Typically, the leather straps are connecting the stirrups to the saddle and protecting the rider's legs from sweat. (Stirrup is a light frame or ring that holds the foot of a rider, attached to the saddle by a strap.)
Weymouth is a type of bridle and bit, not a halter. A weymouth bridle is also called a double bridle it is used on highly trained horses that preform advanced and precise maneuver's. The Weymouth bridle has the same basic pieces as a snaffle bridle, Crownpiece, browband, cheek pieces, noseband hanger, throatlatch, cavesson, noseband, plus a second set of cheek pieces that hold the second bit, this can be called a bridoon hanger or bridoon sliphead. You'll also have a curb rein and a bridoon rein along with a bridoon bit and a weymouth bit.
Purple
There are the noseband, the head piece, the cheek piece, the bit, the throat lash, the reins and brow band. Sometimes you get flash bands. Take note that the western bridle doesn't have a noseband. Some don't have throat latches, and a few don't even have browbands.