wikipedia defines them best in short. Your best bet is to get a good book on equitation and enlist the help of a safe and knowledgeable coach. Once you read up on the subject of jumping you can pick the brain of a potential coach to ensure a proper match. With experience, riders learn by feel to instinctively choose the correct release that works for them and their horse, personally a long crest release best suites my favorite mare as she is long necked and jumps with a lot of power. Look into them all, ask a local school if you can watch some of their riders and it will help you a lot with your schooling. Hope this helps.
Types of releases
Difference releases are used depending on circumstances. However, all riders should strive to readily be able to use each one as needed, and should practice them all.
Basic release: the rider grabs the mane of the horse. This allows the rider to avoid falling back and hitting the horse in the mouth or back, or try to use the mouth for balance. It is used by beginning riders who do not yet have the position and balance to do more advanced releases without risking hitting their horse in the mouth. It is also used by more advanced riders on green horses, who tend to overjump, or when a rider for some reason loses position due to an awkward jump. Riders should not think it is reserved for novices, and all good riders will grab mane to save their horse's mouth. However, it provides the least amount of control. It should therefore not be used if it is not required.
Short crest release: the rider slides the hands up the crest as the horse takes off, not before (which "drops" the horse). The hands should not slide far up the crest, only a couple inches, as needed. It provides support for the rider's upper body, while still providing a good amount of control due to the fact that the rider did not release any more than needed. Best used on verticals, when the rider needs to turn mid-air, or when going down drop fences. An intermediate release.
Long crest release: Similar to the short crest release, in that the rider slides his hands up the crest, but the hands are pushed much further along the neck. Gives a great deal of freedom, but fairly little control. Best for very wide oxers, to allow the horse to really stretch across, or for green horses that may jump large or awkwardly, for gymnastics grids, and for use on horses that have been hit in the mouth over fences and are reluctant to jump or stretch down over a fence. The rider should be careful not to associate a large movement forward with the hands and elbows to mean a movement forward with the hips. The hip angle should still close backward. Many riders get into the habit of jumping ahead with this release. Critics say this release is overused and exaggerated by hunter riders; in the hunter ring it is often used by experienced and/or professional riders on well-trained horses over jumps of relatively modest size. Proponents say that this shows off a talented hunter by proving the horse needs little assistance from the rider to jump in good form.
Automatic release: The most advanced release, where the rider maintains a soft, elastic, steady rein contact with a straight line from elbow to bit as the horse jumps. This release results in a great deal of control and communication between horse and rider, allowing the rider to signal to the horse what to do on landing. It allows the rider to better turn while in the air, to correct horses that jump crookedly, and to provide the support that some horses prefer over fences. It is also good on horses that need lots of control after landing. However, the automatic release requires perfect balance from the rider to be executed correctly. An unsteady rider will catch the horse in the mouth, and an incorrect automatic release will spoil a horse's jump and confidence. However, if a rider is able to perform it correctly, it is often best to use this release whenever possible. Very useful in show jumping and eventing, where control is very important.
Slipping the reins: Not a release, per se, simply letting the reins slide through the hands a bit. It is a very valuable tool all riders should have. To be used when the rider gets behind the motion by accident, when riding drop fences or fences with a downhill landing, or when a horse gets in trouble over a fence (such as catches a leg on a cross-country fence). Allows the rider to release the horse and give him the freedom he needs, without forcing the rider to do one of the other releases (which are inappropriate for drop fences). However, it results in extra long reins on landing, so the rider must be able to immediately gather up the reins and shorten them to the correct length.
Incorrect release: When the rider releases by shoving his hands above the horse's neck, into the air, or way down below the neck. This not only looks sloppy, but changes the rider's balance over fences and you will hit your horse in the mouth upon landing.