There are heaps of different studs around and each of these studs are made for different conditions. Below are a number of the more basic studs that are avaliable.
Keepers are designed to keep the stud hole clean and undamaged when the horse is turned out in the paddock or in the stable. There are three main types of keepers avaliable in New Zealand (as always there are more avaliable overseas). Metal keepers (Pictured), Rubber Stoppers, and Stromsholm Keepers. I prefer the metal keepers as they are simple to take in and out. Rubber plugs need to be pried out with something sharp and Stromsholm Keepers use Alan Keys to get them in and out. Some people also prefer to plug the hole with cotton wool.
Road studs are usually used on roads or on very hard ground. Road studs can be used on the front or back and on the inside and outside of the shoe. Generally road studs are the best studs to use on the inside of the shoe as they are blunter then other studs and are less likely to cause damage if the horse should stand on itself.
Grass studs are avaliable in a few different sizes. Grass studs are longer and generally narrower than road studs so they can dig into hard, dry ground. Contrary to the name they don't have to be used everytime you ride on grass, road studs can be used too. Grass studs should be used the when the ground dries up and gets hard, which can cause your horse to slide on the slick grass. They can also be used when it has rained on hard ground, causing the surface of the ground to be really slippery.
Mud studs are used for very wet and soft riding conditions where deep traction is needed. Some mud studs are called Olympic Studs and these are used for extremely slippery ground. Generally, small mud studs are used in the front shoes and slightly longer mud studs are used in the back shoes when conditions are wet and slippery.