Horseshoe studs are small studs applied to the bottom of horseshoes to increase traction. These are different from the small 'heels' that often appear on the ends of horseshoes; these are called caulkins, and also serve to improve traction, but caulkins are usually part of the cast shoe and cannot be removed, while studs are usually screwed into to the shoes and are removed after every ride.
Studs are commonly used in horse trials, polo, and show jumping, and may be used for any ride during which you expect to encounter terrain that the horse may slide or slip on. They do require specially prepared horseshoes in which a farrier has drilled specifically sized holes into the shoe for the studs to screw into.
There are a lot of specifics to using studs; studs should be balanced on the left and right side of the shoes, but sharp studs can be dangerous, so the inside studs are often duller that the outer studs. Horses also will often wear protective leg boots so that they don't accidently scratch their legs with the studs. Studs do affect a horses' gait, so horse should be given a chance to adjust to wearing them, and they should be removed when a horse is at pasture, in stable, or on terrain that does not require them.
There are dozens of studs available for purchase, but there are three traditional types of stud that most often referred to. Each of these types come in various sizes and shapes, and may be referred to by more specialized names, but the basic forms are:
Road studs or flat studs are used for extra traction on hard-packed or paved surfaces. They look a lot like normal bolt heads with a small point or peak, and are the most low-profile of common studs.
Grass studs are longer, pointed, and look a lot like football cleats. These are the sharpest of the common studs.
Mud studs or deep footing studs are larger, but less pointed than grass studs. They are designed to descend through the top layer of soft mud, but do not need much piercing power.
Additionally, there is the polo stud, a stud that meets the official regulations set forth by the governing bodies of polo, and the dressage stud, which is essentially a small grass stud.
If you are considering using studs you should check with your vet, and make sure that your horse is healthy and does not have any hoof or leg conditions that would be aggravated by changes in gait resulting from stud use.
Horseshoe studs are sometimes referred to as horseshoe cleats, although this term is more often used to refer to a horseshoe-shaped piece of metal with attached cleats that is affixed to the heel of high-heeled (human) boots.