Reddit mentions of The New Encyclopedia of the Horse

https://www.amazon.com/Encyclopedia-Horse-Elwyn-Hartley-Edwards/dp/0751312363/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=&sr=

A used copy of the Encyclopedia of the Horse is probably going to be best bet for getting a quick and dirty general reference for the horse wild in general. There are some other books, but they are largely geared toward people who are more personally interested in doing things themselves.

That said, google and wikipedia is generally your friend.

Also, people here are always happy to answer your questions.

It sounds like your daughters are riding at what is called a "Hunter/Jumper Barn." This refers broadly to the two main disciplines that involve riders jumping horses over obstacles in an arena or stadium. The obstacles are called fences, and the fences are made in a way that they fall down easily if the horse hits them.

"Jumpers" is an easier event for the layperson to understand. A jumper competition is one where the riders compete to see who can ride their horse through a course of fences the fastest and with the fewest number of "faults" (a fault is something like going off course, knocking down a jump, or taking too long to complete the course). So, the winner is determined objectively- whoever gets the best time with the fewest faults wins. Jumpers descends from military riding competitions, but is several steps removed from that.

"Hunters" takes its name from the practice of fox hunting from horseback, but it share almost nothing with actual fox hunting. The trappings of the hunters and the jumpers is similar, but there are some significant differences. Hunters is a subjective event. There's also a predetermined course for the rider to complete, along with an ideal time and a penalty for faults, but the rider is graded subjectively by a judge.

Broadly judge scores the riders based on their form, their control and precision as they ride to the fences, the relaxation and willingness of the horse, and the overall quality of the ride. The riders are scored according to the subjective grading of the judge (who has standards they judge by, but it's still in the opinion of the judge).

So, the jumpers is about speed while the hunters is about presenting good form. Most barns that specialize in jumping will offer both disciplines, but some are more specialized than others. Generally, younger students will start out as hunters because the fences are lower and the classes are more about presenting correct fundamentals, which makes it a safer event for beginners showing for the first time. The courses also tend to be much less complex and are designed to allow the rider the best opportunity to set their horse up correctly for the fence.

The jumpers is faster and arguably more athletic. The courses are more complex and require more advanced tactical thinking from the rider. It's a more aggressive show ring and one that requires a certain level of maturity, because a bad decision on a jumper course is more likely to result in some level of injury for the horse or rider. Now, injuries are uncommon, but there are EMTs at big shows for a reason...

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There are some other "English" events as well. Dressage is a discipline that does not involve jumps. Like the hunters, dressage is about creating a nice image with correct form and presenting that to a judge, who scores and grades the riders and horses as they complete a prescribed series of movements, called a "Test." The tests range in complexity from basic walking and trotting in a few circles around the arena to preforming highly complex movements and patterns at the Olympic levels. Each movement in the test has a maximum score of 10 points, and the horse and rider team is scored a 1 to a 10 for the execution of the movement, with 6 being about average but acceptable to 8 being about as good at the average horse and rider could hope to accomplish. A rider getting a 9 or a 10 on a movement is something that people will talk about for weeks, and a horse getting a "10" on a movement at a recognized show will add at least a thousand dollars to its value. The scores are added up to a total, and the riders score is expressed as a percentage of the total possible points possible for the ride. So, a 60% is an acceptable score, while a 70% is a good score and an 80% would be an excellent score (a score in the 90s would be incredible).

Think of dressage as something like gymnastic floor exercises. However, dressage is really about the athletic development of the horse through proper riding and conditioning. It's about teaching the horse to use his body in the most bio-mechanically correct way. In that way, dressage is like equestrian weight lifting- its the proper use of form while lifting to develop an ideal body, while also improving the quality of the horse's movement to its best possible state.

Dressage is also a descendent of military training. Dressage riding and training was something done by cavalry officers tasked with training horses for drill and combat, with the purpose of making horses strong and obedient. However, as horses became less useful in combat, dressage became more of a military art and less about practical combat training. Dressage can perhaps be compared to military drill, where the Marines silently perform a routine with the utmost precision. But, it's also been separated from the military nature of things by many years and has become more of a sport or a game, which is a source of some controversy. But, that's also getting outside of the scope of an explanation for a layperson.

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3 Day eventing is the other main English riding discipline. 3 Day eventing is also a military inspired event and it retains more of its military heritage than some of the other events. As the name implies, a 3 day event lasts three days and consists of three events- jumpers, dressage, and cross country. Jumpers and dressage work exactly the same as the other two events that I described, but the dressage tests tend to be less complicated.

Cross country is the real reason why anybody events. Cross country is the ultimate test of a horse that can jump. A moderately difficult cross country course is enough to make an experienced grand prix Jumper or Hunter nervous. Cross country is about galloping at speed over terrain. The jumps are fixed and do not fall down if hit. There are hills, slopes, banks, and water complexes that the horse and rider must navigate. Scoring is based on time verses faults, like jumping. The winner of the event is determined by the overall ranking in all three events.

This event has more fatalities attached to it than almost any other horse sport. There's usually at least one horse and rider fatality per year, if not more. I think over 50 riders have died since 1990. To do this event at the competitive level, you have to be equal parts skilled and crazy. Bad injuries can happen even at the lower levels. There have been some safety improvements, but you rarely see eventers under the age of 15.

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This is already a novel, so I'll end here. If you have more specific questions, I'm happy to help answer them. I can help explain more about the horses themselves, breeds, health, general training, attire, bits, saddles, other equipment, and what to look out for when dealing with horse trainers (who are, in general, money grabbing liars who are looking for ignorant parents with horse crazy daughters that they can exploit).