While stargazing, I noticed a large circular band of light surrounding the moon.

Does it indicate that there is unsettled weather on its way?  More snow or rain?

Folklore would say so, but in fact it is an indication that there is a layer of microscopic ice crystals high in the atmospherewhere incoming light rays from the moon are bent, (diffracted) by these ice crystals at an angle of 22 degrees.

So you are seeing dired moonlight and also seing diffracted moonlight appearing as a circle  of 22 degrees away from the moon.

It’s a glorious sight

.circle of light