Moon Dogs ,or 'mock moons' are the analog to sun dogs. When the Moon or the Sun is observed through a cloud deck that contains icy particles such as cirrus clouds, the icy particles can serve as a screen and diffract the light from the Sun of Moon into a variety of wonderful shapes. Most of us have seen lunar haloes or haloes around the Sun. In the far northern latitudes near the arctic circle, spectacular halos and other 'glories' can be seen. These can be quite complex with multiple concentric circles, arcs portions of arcs, all arrayed in spectacular geometric designs. Sometimes if the clouds are clumpy enough, but not so great as to destroy the careful optical conditions, you will see bright spots in the sky called 'dogs' which look almost as bright as the Sun or the Moon itself and are situated on a ring surrounding the Sun or Moon, sometimes on opposite sides of the real Sun/Moon. The angular separation is usually 22 or 46 degrees. Sometimes the ring itself will be faint, giving the illusion that there are two or more Suns or Moons in the sky.
Presumably all of this is controlled by the shape of the ice crystals in the clouds, their altitude and deployment in the sky, and the laws of optics. I am pretty comfortable with such an explanation rooted in physics, but I still get goose bumps whenever I see such things. It's nice to know that the real world holds within it unexpected phenomena which are 'perfectly natural', yet like a rain bow, glorious to behold.