Camber - "A slightly arched surface." -Dictionary.com. If a longboard has camber, the middle of the board will be higher than the nose and tail. One way to imagine it is if you flipped the board over (trucks facing up) you could rock the board back and forth instead of it sitting flat.

Camber is good for absorbing bumps in the road. It also helps absorb the energy that is released coming out of a turn when pumping or carving. Many slalom boards have camber. The camber of a board will eventually, lose its spring. Pumpkin longboards are famous for camber.

Flex- A board is said to flex when it bends some under weight. Boards with camber are meant to flex down some and then spring back up. Longboards with a notable amount of flex are Flexdex, Fibreflex and Supaflex.

Concave - This means that the deck slopes down from the edges towards the center. A concave deck is usually stiffer. Usually a concave deck does not have camber, but there are exceptions.

Concave decks are good for tricks as they help your feet stay on the board.

Convex - The opposite of concave. Very few (if any) longboards are convex.

ADDITION:
It has been brought to our attention by Philippe at Chronic that there is in existence at least one longboard with both convex and concave. It's called the "Golgoth." Below are a couple images.

ADDITION 2:
Ben says:: One of your pages reads "The camber of a board will eventually, lose it's spring. "

Maybe a low-quality soggy wooden ride will lose its camber, but the at least my Pumpkin hasn't lost any camber at all.

I think it's negligible, and isn't worth mentioning.