On-Axis lights are used to illuminate reflective objects.

The mail problem with imaging reflective objects is that reflections of the light and camera can often be seen in the resulting image.  On Axis illumination uses beam splitting techniques to enable the illumination to be positioned between the camera and the object so that the camera is situated behind the light.

Coaxial Spotlights are used in conjunction with coaxial lenses.  The spotlight looks similar to the end of a fibre optic bundle and plugs into the side of the coaxial lens.  Coaxial lenses generally use prism assemblies inside them as beam splitters.  Thus the illumination is projected from the end of the lens which is often very convenient especially for small fields of view.