The ring around the Moon is caused by the refraction of Moonlight (which is really sunlight) through ice crystals suspended in the upper atmosphere. These ice crystals always have a hexagonal shape, so they always refract light at the same angle. Of course, the atmosphere is filled with crystals, all refracting moonlight off in different directions. But at any moment, a huge number happen to be in just the right position to be refracting light towards your eyes. You just aren’t in a position to see all the other refracted light. This is what causes the Moon ring.

We see a rainbow because the different colors are refracted at slightly different angles. This is exactly what happens with a rainbow. The moonlight is broken up into its separate colors because they all refract at different angles, and so you see the colors split up like a rainbow.

A moon ring or moon halo will always be roughly the same size – 22-degrees across the sky. The full moon is half a degree, so the ring around the Moon will encompass an area 44 times bigger than the Moon. This is because the crystals bend the light at a 22-degree angle from their original path.

This is the same principle of why we can sometimes see a ring around the Sun.

This site has a great description of why we see the 22-degree halo.

A moon ring, also known as a winter halo, is a phenomenon that usually appears in conjunction with a full moon. There appears to be a whitish ring, approximately 10 to 20 times the size of the moon, surrounding the moon and centered on it. It is caused by refraction of the light from the full moon in the ice particles floating in the clouds, as opposed to a rainbow, where light refracts in the water vapor that makes up the clouds. Since this happens most effectively at a certain angle, this ring appears at the bottom of the clouds, and since similar triangles must form between the moon, the refracting surface, and the observation point, the "highlighted" clouds are at approximately the same distance from the moon, creating the image of a ring.

In ancient beliefs it is believed that a moon ring means very warm days before the winter storm. It is also said that the number of days can be counted by counting the stars inside the ring. If you count 15 stars then you have 15 days before a winter storm is to come.