The concave is designed into the shape of the handboard based on it's performance in different kinds of waves.
The best example of concave is the Bulaboard, which was shaped over a mayonnaise jar by world famous bodysurfer Sean Enoka. The steep concave is designed for fast, steep, shorebreak. In a nutshell the concave acts like a fin on a surfboard, with a deeper concave you can dig into the wave. The concave gives you a hold, almost like a suction cup to the wave, allowing the board to stay and keep the line of the wave a lot better than something flat.
The flatter the bottom of the handboard is, the more likely you'll skim off the top of the wave. However, the width and length of the board, along with the concave of the handboard also factor into the performance.
The Wedge board is a bigger board designed for everything, but shorebreak. The slight concave down the entire length of the board, channels water down the center for less spay back and a little suction on the wave. This allows for the board awesome ability to do turns, cutbacks and all sorts of fun tricks.
While, the soft-top Gromboard is similar to the Wedgeboard, with a slight concave down the center. However, the soft-foam material makes for extra buoyancy making it an all around fun board, while being less speed driven.
Lastly the Phishboard has next to no concave and looks like bullet because it is so aqua dynamic, due to the long length of the board and fishtail there is no need for a deep concave.