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:
