What is a shadow ray? I have read the Blender documentation wikis on both the light path node and on light paths.  But neither of them seems to clearly explain what a shadow ray is.

For instance, the light paths wiki says:  


  Shadow: the ray is used for (transparent) shadows


It then shows a diagram with the ray labeled “shadow” going (backwards) from a surface to a light source.

This doesn't make much sense to me.  The text doesn't make sense because in cycles (as far as I know) a shadow is just an area with less light than the areas around it, which is not a technical value.

From the diagram, it looks like a shadow ray is a ray that is about to hit a light source (speaking from the actual backwards ray-tracing point of view).  But I don't understand how that works either, how can cycles know what a ray is about to hit?  Especially if that information could be used to change the direction of the ray.

So what is a shadow ray?
Shadow ray is the last segment on the light path traveling through the scene (from camera to light source). It is just a technical term, only a label. In nutshell it marks anything that goes to light.

When the light path begins, it starts as a camera ray, when its reflected it becomes reflection ray, when it goes through glass it becomes transmission ray and from the last point to lamp it is shadow ray. These are the base 4 types of rays in cycles (they divide more into diffuse, glossy etc. also).

With Light Path node and Shadow rays, you can for example color shadows to some tint of your liking. Here the object will be transparent and blue for the shadow part of path but opaque and green for the rest of it:



Note: In 2.75 rendering colored shadows from transparent objects is buggy - when adding Transparent BSDF in rendered viewport the shadow won't be colored - you need to restart the render-viewport (on CPU and GPU).