Breast collars, cruppers, and bridles come in nylon, leather, and Bioplastics. Ordinary nylon gets stiff from absorbing salt and dirt, and is hard to keep clean. It is available in many widths and colors, and more and more suppliers are making horse tack out of it. Traditional leather can cause chafing and will need regular care to stay soft and pliable. However, leather will break in case you have a fall, and this is a good safety feature.
Serious accidents can result from the rider being hung in the tack.
A good breast collar can help hold the saddle in place, preventing it's sliding back or roiling sideways. These don't do much to prevent the saddle from shifting sideways, either. This works much better without soring the horse. If you decide to use one, be sure to accustom your horse to it at home. In fact it's not a good idea to try out any new equipment at a ride.