It is not a single projectile such as a bullet but rather something similar to bird shot.
I have been doing Cowboy Mounted Shooting for about 2.5 years. My wife also does it. We actually just started our own club in the area. It is tremendous fun and most horses take to it better than you would think. It is important to hook up with a good club and properly introduce your horse to it.
I did want to respond to the above comment. There is NO projectile involved in mounted shooting. All three organizations - CMSA, SASS and MSA - use basically the same ammo, and none contain projectiles.
The ammo used is a black powder blank. Black power is the "old" style gunpowder that makes a bigger noise and lots of smoke. (As opposed to modern ammunition which uses smokeless gunpowder).
The cartridge casing is filled with gunpowder and then typically crimped to hold the powder in.
When the gun is fired, the black powder ignites. The burning embers of powder ejecting from the barrel are hot and enough to pop the balloon. I guess you could nitpick and call this burning powder "projectiles" but it is not. If you look at pictures on the CMSA website, you can see the flame of the burning powder coming out of the barrel and also the smoke trails of burning powder.
Many people are under the false impression that we use some type of projectile like cornmeal. That is definitely not the case. This sport realizes the potential for danger and has some very strict rules about safety. For example, no live ammo is EVER allowed at a shoot. All shoot ammo must be provided by the club and be purchased from a certified supplier.
As someone mentioned, the burning powder discharge is enough to pop a balloon at about 20 feet. The safety distance from the edge of the arena is calculated with this in mind.