Phong shading was invented by a Mr. Wu Tong Phong. It improves somewhat on the Gourad approximation for shading polygons. The basic principle is to calculate the surface normal at each vertex of a polygon, and interpolate the vector over the surface polygon. Then, for each pixel, calculate the brightness of the pixel based on this normal vector.
This is a very slow process indeed, and one which cannot yet be considered a realtime option for home computers. Many programs which claim to be able to perform realtime Phong shading are infact using fake phong shading. This is very nearly as good, and is very much faster.
This is a very popular method for shading polygons, especially in non-realtime rendering packages like 3D Studio. However, although it looks much nicer than Gouraud shading, it is still not physically accurate.
To gain a good understanding, you should also read the article entitled 'A Physical Model of Light'.
Phong shading is just an extension of the Physical Model of Light in the previous article. Every single pixel has it's brightness carefully calculated using the interpolated normal vector.
In reality, very few materials exhibit this ideal property. Microscopically rough surfaces, such as corsely sanded wood may come close. Most surfaces also exhibit specular reflection to some extent. Phong shading does not deal with this. It also does not handle the fact that brightness is inversely proportional to the inverse square of the distance from the light.
Now, the dot product function returns values in the range 1 to -1. 1 is full brightness. There is no such thing as negative light, so values less then 0 should be taken as 0. If you multiply this value by the brightness of the light, then you have the brightness of that pixel.
I am not going to bother to demonstrate this algorithm or give any pseudo code. Those who are smart enough to implement this really don't need my help. People who really want Phong shading in their programs should read the next article on Fake Phong shading.