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.
Slack line cast
This is primarily a dry fly cast. It is also called a "wiggle" or a "serpentine"cast. Like several other cast, it puts slack line on the water and this gives you a longer drag free drift over a rising fish. It is very good when you have variable currents between you and the fish.This is a pretty easy cast to learn.
The pile cast (often called a parachute or puddle cast) is very useful when you are above a fish. With this cast you in effect "pile" up the end of the leader and tippet to get a longer drag free drift over your target. I have also found it to be very useful when fishing back eddies.
Pile Cast - Bumcast (YouTube)
The Parachute and Pile Casts - Orvis, Pete Kutzer (YouTube)
The reach cast or "aerial mend" cast is very useful in many situations where you are casting across different currents and need to get a longer drag-free drift. It is a traditional overhead cast and a mend in a single casting stroke. I find that the reach cast also enables a better or longer mend and it does not disturb the surface as much as a traditional mend.
The curve cast bends either to the right or left of you and is a "hybrid" of the standard overhead cast. Curve casts are used when you need to cast the fly around objects on the surface or when you need to prevent the leader and fly line from "spooking" a fish as the fly passes over the fish.