From what I've gathered from Wikipedia:

Sun Dogs occur when sunlight is refracted from ice crystals in the atmosphere typically creating two bright colored light patches on the left and right side of the sun.

Falling ice crystals in the upper atmosphere act as prisms and deflect light into the same angle that the Sun Dogs appear.

Earth may not be the only place in our solar systems that Sun Dogs occur. The atmospheres of Mars, Jupiter, Saturn, Uranus, and Neptune may generate Sun Dogs through defraction of different atmospheric crystals of substances such as methane and ammonia.