The same thing during the day is called a rainbow. It works the same way as its daytime counterpart, it just uses moonlight instead of sunlight. Any rainbow comes from light hitting drops of rain and going into the drop and being reflected off of the inside edge. As the light leaves the drop on it's return journey the other side of the drop acts like a prism and separates it into the colors of the rainbow. Daytime rainbows work great because of the intensity of the sun light, but with moon light it makes something shaped like a rainbow with just a hint color.
Moon bows occur in the side of the sky opposite the moon.
Visit Matt's Homepage for more great photographs. Atmospheric Optics - A site with lots of good technical info and illustrations. Light Pillars - An explanation of how light pillars are formed. Moon Watch - A question and answer page about moon halos. This site explains how and why we see halos.