Equestrian photography is one of the lesser known, though nonetheless still popular forms of photography. For those of you who do not know what it is, Equestrian Photography is the Photography of Horses, both in their natural surroundings and in competitive situations.
Equestrian Photography can be divided into the following sections:
1.  Competitive Equestrian Photography
While this gives one visions of photographers jostling each other out of the way to get the best shot, I am merely referring to the practice of photographing horse sports!
Horse Sports include the following disciplines:

Racing
Showjumping (known in America as stadium jumping)
Dressage
Showing
Eventing
Equitation
Driving
Vaulting
Western Pleasure
Western Trail
Barrel Racing
Rodeo
Gymkhana
Polo
Polo-cross

There could be others, but these are all the ones that I can think of right now, in mainstream equestrian competition.
What is the Key to Being a Successful Equestrian Photographer?
The key to being a successful Equestrian Photographer, is that you must know your subject, the way portrait and wedding photographers, know theirs.
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Sony World Photography Awards is one of the most comprehensive photographic competitions in the world. The shortlist for the 2010 global program for both professional and amateur photographers wasThe way that fashion photographers understand models. That, in a nutshell, means that you should be a horse person. You should understand horses. You need to understand the way a horse moves.
Your success as an equestrian photographer is absolutely dependent on being able to capture a horse in their most classically correct positions, and make all the riders look like top athletes.
Let us take Dressage as an example. Dressage is a highly specialized form of horse sport. Each movement requires the utmost precision by the rider, and years of training for the horse and rider to get it right. 
I was once in a line-up of three photographers at a Dressage show. Two of us, knew each other from the local equestrian circuit, and were enjoying the spectacle, animatedly chatting away about the new faces, new horses, and how the competitors and our regular clients were doing. The third one in our group was new. We did everything we could to help her, and explain what was going on etc, and while she valiantly snapped away, and tried to follow our advice about how to best capture the horses, at one point she exclaimed, “But this is like watching PAINT DRY!”
She had absolutely NO idea about how best to capture the horses, because she did not understand the sport.
Time and experience will teach you how to take good photographs, but only passion and interest in your subject will make your photographs truly great. Since making money by capturing equestrian competition, is largely dependent on selling your photographs to the competitors (sadly newspapers often have bigger sports in mind), you need to understand the sport, to be able to make your competitors look good.
This means knowing what angle to capture each movement at. It often, in the case of dressage, means memorizing the test being performed, so that you know exactly which move is coming and when, so that you can be prepared for the ‘extended trot’ for example, and be sure to get at least a few great shots of that.
The better you can make your competitors look (in other words the closer they look to that image of perfection seen in the very top riders) the more likely they are to buy your photographs. 
A fellow equestrian photographer and I always joke and talk about our best shots in terms of the size they sold for, in other words, the better the horse and rider combination looks, the larger the print they ordered of it.
I have seen many ‘wannabe’ equestrian photographers. They come and they go. Those without real passion and conviction for their subject, DO NOT LAST. The photographer I mentioned who talked about it being like ‘paint drying’ lasted just–one–day.
However, do not be despondent, if there is one thing that horsey people like nearly as much as horses, it’s talking about horses. If you don’t know what you’re looking for when you start out, ASK.
Meet people, pat their horses, ask about their breeding, their training, and learn as much as you can. People will be very happy to share information with you, and how they want their horses to look, so there is always hope, if this is the avenue of photography that you wish to pursue.
2.  Natural Equestrian Photography
This area of equestrian photography usually arises out of the competitive side. That is, clients who purchase your photographs of their competitions, will sooner or later ask you to come and take photographs of their equines in their natural state at home. While this draws many parallels with competitive photography, in that you need to understand (or at the very least not be afraid of) horses, it is a different ball game.
It is usually best to charge, as you would for a ‘human’ portrait session. Chances are your subject is going to include the rider anyway, whether in their riding gear or not. Usually the natural side of it will include numerous head shots of the horse with or without the owner/rider, and also shots of the horse running free. There are many techniques one could employ to get your subject moving, and getting the best out of them, but this is not the place to go into those lengthy explanations.
Lastly, to be be a good equestrian photographer, you need to have the following personality attributes.

You need to love horses.
You need to love standing, a lot. Like for 8 hours a day. You will be the first to arrive at the show, and the last to leave.
You need to be prepared for all types of weather. You will get rained on more than once, I can assure you.
You need to love fairground food (hot dogs/hamburgers) or be disciplined enough at packing sandwiches.
Horse poop is basically just grass, so when you stand in it do not scream like a girl. It is okay. You should be wearing boots anyway.
You need the bladder control of an elephant! If you want to get all the shots, then you *can’t* go when you want to. Hope for a lull in proceedings and then run like hell.

Smile and make friends with other equestrian photographers, and HELP each other. Chances are you’re the same kind of person, and will actually become life long friends.