Pole beans have an
indeterminate growing habit. They need a 6-8 foot tall trellis to climb. They will give 2-3 crops over an extended period of time.
Bush beans have a
determinate growing habit. They are generally self-supporting "tree" type plants. They will give one big crop and then a slowly declining smaller crop over a relatively short period.
Originally, the majority of beans in home gardens were pole beans because of the gradual crop that was good for several meals. Then farming switched to bush beans which were ideal for machine harvesting. Now, most beans grown in home gardens are bush beans.
It has been said that there is an bush equivalent for nearly every pole variety and vice versa.
Snap beans are simply any bean like a green bean that you pick when the seeds inside the pods are not "bulging" or mature.
Shelling beans are any beans where you let the seeds grow to maturity (noticeably bulging inside the pod or even until the pod is dried out and papery) and discard the pod. An example is rattlesnake or kidney beans where the beans themselves are important but the pod is not. Some beans are good as both snap beans and shelling beans (Dragon Tongue), some are better as snap beans (Fortex, Contender), and best for shelling (Rattlesnake).
I think drying beans is just another word for shelling beans.