Sidewall (sandwich) The most used nowadays, you may not even know there are others, that's because the sidewall construction transmits more energy to the edges, in a more direct way, minimizing energy loss. This means that you get a bit more control, the board gets more responsive, grips more. Also, you can feel the snow better, and It looks good too.
Half cap construction Something between the cap construction and the sidewall. It's used specially combined with the sidewall construction. Some higher end boards may use, for example, half cap on the tips and sidewall in the midle of the board's efective edge.
Cap construction Power transmission is more diffuse, which translates into less edge catching and a more playful ride. Good for kids and beginners.
Slim-Tip construction Slimmer tips will make the board better to rotate and easier to turn. They look good too.