The Bayer sensor used by the vast majority of cameras is basically a two-by-two grid of sensors with 1 blue, 1 red, and 2 green sensors known as a Bayer filter named after the Kodak Labs scientist that came up with it. The data from such a sensor then must go through a demosaicing process that converts the 4 data points into a pixel giving the result of the 3 color merge. The reason for 2 green sites is that the human eye is reported to be more sensitive to green and so the color is emphasized in the system.

The Foveon model, which totally fascinates me, is an approach to follow a more traditional film style. In this context, the idea is that the three primary bands of light operate at different wavelengths and so penetrate the sensor material to different depths, the premise of color film. In this case, blue is the least penetrating and red the most, so by stacking the layers, they can detect at each photo site the level of each of the primary colors. The technology, as a result, eliminates the moire pattern than can result from the demosaicing algorithms associated with a Bayer filter and give a more accurate result.

I'm really excited about the Foveon technology and I'm looking forward to seeing where Sigma takes it. They've finally produced and APS-C camera with this sensor, so when the reviews and samples finally hit, I'm going to be looking at them closely. Having said that, I think the camera makers have done a very good job with the Bayer model, it's a proven and well-understood means of image capture and that can be seen from the often stunning results. If the Foveon exceeds that, we're in photography nirvana. :)

Anyways, I linked some relevant Wiki articles on the two which I think will really help you see the differences.