Hunters, equitation, and jumpers: they can easily all look the same if you don't know exactly what you're looking at. It all just looks like horses jumping over poles, right? The three disciplines are actually very different one another though. Some horses will excel in one single discipline, while sometimes there is an extremely handy horse than can successfully compete in all three! The three styles are easily separated as hunters are focused on the horse, equitation is focused on the rider, and jumpers are all about speed.
Hunters is the most 'basic' of the three disciplines. The jumps and obstacles are usually very conservative and natural, and usually only include colors like white, brown, and green. They are meant to mimic natural obstacles you would encounter out on a foxhunt. A hunter course will usually consist of 8 to 12 fences, arranged in a basic pattern including simple outside and diagonal lines.
In a hunter course, all eye are on your horse. For a nice hunter round, you want your horse to go around at a nice consistent speed. You don't want a peanut roller, but you also don't want the speed you're looking for in jumpers. The horse should be calm and easily controlled. Are the horse's knees even, do they find the right striding to the jump, stay quiet throughout the course, and have nice lead changes? Do they look comfortable and pleasant to ride? Your tack for a hunter class should also be very conservative. No boots or wraps on the horse, a simple snaffle bit, and traditional bridle. Riders should wear a black helmet, black gloves, tan breeches, a belt to match your tack, black field boots, a white show shirt, and a dark colored hunt coat such as navy.
Equitation is focused on the other side of things - the rider. You still need a nice consistent horse for the equitation, but rather than having a ride that does what it takes to make the horse look good, an equitation horse needs to be able to do their job and make the rider look good. No back cracking jumps and crazy bascules here, or else it would be hard for you to equitate well over the jumps. Equitation courses are meant to test the rider, so the course will a little more technical than a hunter course. The jumps will have a little more personality to them, and the course won't be as simple. There will be technical challenges for the rider, which will increase in difficulty depending on the jump height and level of the class. These can include rollbacks, fences on a figure-8, or trot jumps. Boots or leg wraps can be used on your horse's legs in an equitation class, and bits can be a little less traditional, but still simple, such as a pelham. Riders can also be a little more "colorful" with their outfit, by wearing a lightly colored show shirt, and different colored jackets such as gray or brown.
Jumpers on the other hand, isn't a beauty pageant for the horse or the rider. It's all about keeping all of the jumps standing, in the shortest amount of time. If you go over the allowed time for the course, or you knock any rails down, you'll get faults added to your final time. Fences in jumpers will be bright, colorful, and lofty jumps. The course will often have a sporadic pattern and will be very technical. While you still have to have regard for safety and correct riding, in this class it doesn't matter if you have a leg like cement, or a horse that has his knees up to his ears over a fence. Most any kind of tack can be used - as long as it complies with the show's rule book - and saddle pads and ear bonnets can be fun and colorful.