While surfing, what do the different concave types mean? How does it effect you in the water and how the board operates?

Concaves can serve two purposes. They speed to flow of water along the bottom of the board and they allow the board to hold the rail more. The down side is that the deeper the concave the move difficult it is to go from rial to rail. This is often why you will see a double concave in the rear where that part is also slightly veed, to allow the rail to release better.