Whips are: an extension of your arm, a part of your "voice", an aid (as in cuing your horse as to what you want to do. In riding -your hands, voice, legs, seat and back/hips are all considered "Aids". In driving - you have your voice, your hands and your whip), a "bad-boy" correction tool when needed and for me A TRAINING ESSENTIAL.
They are not used to whip or abuse the horse.
As to types - go out to Dover Saddlery, Jeffers Equine, Valley Vet to see what the different whips look like.
a "bat" or a "crop" - is a short handled whip generally carried by an english rider. On the end is often a popper that can make a soft or loud, obnoxious noise. Can be carried by a child and is also short enough to use on the shoulder to help with cuing. Generally, when a child loses their temper - it won't hurt (but yes, it can and a good instructor/parent teaches them other ways). Also used by "gamers" - gymkhana or speed event riders (barrels, pole bending, flag racing, key hole, rescue race etc) - easy to switch hands w/ it and/or can be carried in the mouth while two hands used for direction.
A "dressage whip" - comes in several lengths and was originally designed to be carried by a dressage rider - ability to reach behind their leg to enforce a leg cue. I like them for working on the ground and had one I used a lot while driving our smallest shetland - no "real" lash but that was ok. Can have a sting (not a bad thing) or be mild - depending on flexibility, length and force used.
A "Quirt" - Spanish word derivative. A short, flexible whip with split "falls" (long tassels) usually braided from rawhide and leather. Was carried by vaqueros to make cattle move out. Was often attached to the closed reins they used. Now seen as part of the "Romal Reins" in western show classes - primarily Arabian and some of the working cattle events. Now braided with paracord and often seen with speed eventers (see above).
"Over and under" - a short rope often tied to the Speed Event riders saddle horn. Now often seen braided to match their riding clothes/tack. Can't be dropped, but I feel like it's way overused...
"Bull Whip" - used by drivers, generally on the ground, of oxen, cattle in pioneer days. Made with a wooden handle where held and a long, very flexible lash that is held curled into symmetric loops. The good ones took time to make (intricate braiding) and the person handling the whip learned to use it. Has been made popular recently by the Natural Horsemanship 'craze'. Give me one - I can remove my own ear and make no noise!! Limp noodle type of thing, HAHAHAHAH!!
"Lounge/Lunge whip" - usually a lightweight fiberglass rod (about 4' in length) with a rubber handle on one end and a 2' to 6' "lash" on the other end. Used while working the horse on the lunge...
"Driving whips" - seem to be more varied than any other type. The handle can be made out of many different products and the "body" is generally a fiberglass core. They can be different lengths and a lash drop(also different lengths). The "lash" is often used to touch the horse with - to get a bending OR a moving over. Again - the style, length, weight are personal preference while working your horse and driving them. A well made, "top of the line" one can cost several hundred dollars...
Think I covered most, can't keep my eyes open anymore... Night!!