A lot of the stop is a natural thing for the horse to do.
For example, we teach the stop the same on my Hunter Under Saddle horse as we do on the reiners, but the HUS horse will never do a sliding stop (he may if we put shoes on him and really got into him, but its likely we woudl destory his legs in the process because he isn't built for it)
Breeding, conformation and "Want" are the important things to have, to really achieve thos 20foot+ deep stops (you can do it at a lower level without having some of those things).
We start teaching "Whoa" right from the beginning. The young ones aren't expected to do a sliding stop, but right from the beginning we encourage stopping with their hind end under them. So we say whoa, take out legs off a little bit, and then come in with our hands (on the young ones since they have no idea that just taking the legs off means soething). Then you back almost everytime, at least we do.
Combine that with having a supple horse that can engage their hind end, move their body around, then you just start building that stop up. You put baby sliders on and start repeating the process, and you build from there. A lot of the "slide" will come natural if they are started right.
It's a lot more complicated then I said, but that's about the basics, there are little things you kind of have to feel for (ie. A horse dumping on their front it, bulging their belly here or there, etc) and when you feel these you have to change it up.