There are many different types of competitions that a horse and its rider can compete in. There are all different levels of these competitions, ranging from beginner to Olympic level and quite possibly beyond. The competitions also take place around the world. Competitions can be local or halfway across the world. If you look hard enough, you can find a local competition in your area that is fun and low-key.
Competing in local shows is a great way to experience what a show feels like and what is expected of you before, during, and after the show. I competed in a show at the barn that I ride at, and it was fun. I got a sense of how I was supposed to act during the show and had a great time doing it. I am actually preparing to go to a show in May. It's not as small as the previous one that I went to, but it's still a great way to get my foot in the door and get even more experience as to how shows work. Every bit counts, and I know that my confidence in my riding abilities will grow with each show I go to.
Dressage
If you've ever seen people wearing tophats, black blazers, and white gloves riding horses, chances are that they're doing dressage. Dressage is when riders and horses work together to perform a series of memorized tasks in exact locations. Dressage is about building your horses' flexibility, subtlety, and grace to the point where you can ask it to perform a certain action with the slightest touch and it will do it for you instantly. Horses in dressage must be in tune to the rider's every movement, and perform some complicated yet beautiful maneuvers when done correctly. Dressage horses and riders compete using "tests", which are basically sheets of paper telling you where to go and how to get there properly. Tests are not long, and get more and more complex as your training progresses. Tests are performed by one horse and rider at a time in the ring, and there is a judge that grades it using a standard grading system. Each rider is graded individually, and not against the other competitors.
I am learning dressage at the moment, and every time I watch a dressage tournament on TV, I get even more inspired to do it. I have been riding dressage for about 4 or 5 years now, and I love every moment of it. While it is frustrating at times (getting the horse to do something at the exact letter in the ring isn't easy!), it has its rewards, such as when the horse gets "round", meaning when the head goes down and the neck curves downwards, almost creating an arc. The "round" horse is in tune with your movements and is more willing to do what you ask it to. Another name for a "round" horse is "on the bit".
Typical dressage gear includes white or tan pants, white gloves (for beginners only, so that way the judges can see if you're moving your hands or not), tall black boots, a blazer, and for more experienced riders, a top hat.
Jumping
Jumping is pretty self-explanatory. You simply trot the horse up to a fence and it jumps over it for you. However, the horse requires some preparation for this. You need to aim the horse at the dead center of the fence. If the horse jumps it on an angle, it could hurt the horse or cause it to hurt itself when it lands. You also need to be in what is known as a two-point position. It is where you are slightly elevated above the saddle with your knees in the knee rolls of the saddle. Your reins are held above the horse's withers. If you are leaning on the horse while you are jumping, it throws you and the horse off balance. You also need to be 100% committed to jumping over the fence. My current instructor says that horses can tell when you're unsure of whether you're going to make it over the fence or not. I know that I was nervous one time, and the horse that I was riding wouldn't jump over the fence. A week or so ago, the horse tried to go to the left to avoid the jump (he thought he could pull a fast one on me! ;) ), but I steered him towards the fence and jumped it.
Jumping is also very competitive, and the jumps get higher as you go. I am currently jumping small fences in order to get the feeling of how it is done. I was actually not interested in jumping before I switched barns, but I am glad that my instructor has taught me to jump. I feel that it makes me a more rounded rider, and if I am on a trail ride and I need to jump over a log or downed tree, I will know that I can do it. Jumping, like other things involving horses, is also dangerous. You could fall off or the horse could decide at the last moment to jam its back feet into the ground and refuse to jump the fence, causing you to fly forward and off its back. I have watched people jump at the Sussex County Fair, and I've seen plenty of refusals. There's been a few moments where I've thought, "Is she/he going to stay on?!"
Common jumping gear is tan or white pants, open-front boots, and a hunter-jumper saddle. Optional pieces of tack are a martingale and a figure-8 noseband. (A martingale usually attaches to the girth and goes down the middle of the horse's chest. It helps the horse carry its head in a certain way.) In show jumping, faults (usually 4 points) are deducted if a horse refuses to jump, knocks down a jump, or if the course is not completed in the amount of time set.
Steeplechase
Steeplechase is a form of horse racing. It requires the horse and rider to jump over a series of fences, ditches, and rivers. It is practiced in the United States, Ireland, the United Kingdom, Canada, Australia and France. It usually takes place in a cross-country type racetrack. The jumps are usually long.
The steeplechase originated in Ireland in the 18th century. There are two types of American steeplechase: hurdle and timber. Hurdle races are usually 2-3 miles long, and the horses are trained to jump out of stride. Timber races are usually 3-4 miles long, and horses are trained to jump in an arc.