The western saddle has an upright piece at the front which is called the "horn" or "saddle horn." Horns are seen on western saddles, Australian stock saddles, gaucho saddles and other variants. They are not seen on English type saddles.
In other words, saddle horns are put on saddles used by horsemen who are working cattle. The correct purpose is to wrap the end of a lasso or lariat around so that the horse is holding the captured cow or steer, not the cowboy.
Saddle horns have developed other purposes over the years, and are sometimes called the "Oh shit" handle by serious trail riders. (I.e., yes, people do grab the horn to keep from falling off. English people tend to use the horse's mane or a strap around the neck as a "security handle" and some pony saddles actually have a handle on the front).
Trail riders might also hang things such as a quirt or a water canteen or even a camera from the horn...and thus it sticks around even when not being used for its original purpose.