A satin finished ring is created by brushing the surface of the ring in the same direction which dulls the surface. Rings with a satin finish have a matte appearance and don't reflect light the same way a polished ring does. You can see this displayed on the left cigar band in the photo above.
A polished ring has a shiny, reflective surface making it stand out on your finger. Polished rings are very smooth since they are polished with a fast spinning brush to smooth out the entire surface of the ring. You can see a polished ring on the right cigar band in the photo above.