The dot product is a really handy single return value operation that is applied to two vectors at the same time. The result of the dot product is (in technical terms) the product of the length of both vectors and the cosine of the angle between them.

How can this seemingly awkard combination of three different values be useful?

Well firstly, unless we're retarded, we will use get the dot of vectors that are already in a usable form for the task at hand. For example, if we want to know the angle difference between two vectors, we would normalise the input vectors (or divide the dot result by the lengths of the two vectors, take your pick, the math's all works out the same), then we can do an acos on the return value to find out the angle of difference (although you shouldn't really use angles in professional code ;)

Secondly, sometimes the combination of values is more useful than the separated triplet.

Dot product is really useful in lighting. Doing a dot of the surface normal with the angle of the incoming light source gets us an angle attenuated value (just like you want for diffuse mapping).

When we talk about dot3 bump mapping, we're talking about doing a per pixel dot product of the light vector and the normal in the bump map.

Specular is a little more complicated, but its mostly using the dot of a calculated new lighting normal that is created from the incoming lightl vector plus the product of the dot of the normal and the light vector and 2 and the normal itself. (written often as L+N*2*(L.N)) which is then dotted with the normalised vert to camera vector to give an angle attenuated value for the specular.