Fly Casts by Category/Purpose
The following is a list of fly casts categorized by their primary purpose. At the bottom of each description are several links I find to be especially well done with highly qualified instructors and good quality production.
The Belgian cast is also known as the oval cast or the constant tension cast. It is most useful in two situations. First, in windy conditions where the wind is blowing into your casting arm side and second when you are fishing with a double fly setup or maybe with a split shot or heavy nymph and indicator. In both cases, this cast is meant to help keep the fly(s) from smacking you. It is a popular saltwater cast due to the windy conditions that are common.
These two videos do an excellent job of explaining them.
Heavy Fly Casting with Tim Rajeff - Gink and Gasoline
How To Make A Belgian (Oval) Cast - Fly Casting Lessons - ORVIS
This cast is primarily a saltwater cast. You use it often when you are "sight" fishing for bonefish, tarpon and other species commonly found on the flats. If you have been there, it is often the case when you are standing on the bow of the boat looking for targets. Suddenly, the guide or another person in the boat says "tarpon at 10:00 o'clock". Many times, these fish on the flats are moving very quickly. Properly done, this cast allows you to make a 60-75 foot cast with just a few false casts. This is why they call it the "quick cast".
The Quick Cast - Chris Myers (YouTube)
The Salt Water Quick Cast - Capt. Joel Dickey and Gink and Gasoline (YouTube)
The Quick Cast - Chris Myers