The primary releases used are a crest (sometimes called "long", although this is more old-fashioned) release and an automatic release (also called a "following hand). In George Morris's Hunter Seat Equitation he also details a "short" release, although I think these days that would often just be grouped in with a crest release.
A crest release is when you move your hands to about 1/3 to 1/2 of the way up the horse's neck as they jump, pressing your hands into the crest of the horse's neck. This release is used at all levels of riding and is typically the first release a kid would learn when learning to jump, because you can easily grab mane if you are unseated. This release will give you a little additional security, as you have the extra support of the horse's neck, since you're pressing your hands into the crest. You shouldn't be balanced on your hands, resting your weight on the horse's neck, however. I feel as though that's a common error beginning riders will make. As I said though, you see this release at all levels of riding, from kids starting out all the way up to Olympic riders. It's the most common release in the US.
An automatic release (aka, following hand or jumping out of hand) is when you keep the same contact with the horse's mouth over the jump as you do on the flat, allowing your hand to follow the horse's mouth as he stretches his head and neck over the fence. The rider keeps the same straight line from bit to elbow over the apex of the jump as they would have on the flat using an automatic release. This release allows for a more consistent contact and some would argue a greater level of control, as you're not dropping the contact over the fence and picking it up again on landing. George Morris described this as the most advanced release.