Mars does not have any rings. It does have two moons: Phobos and Deimos.
Mars, along with the other three planets closest to the sun, are "rocky" planets, according to the Hubble Site. Only the four gas giants, Jupiter, Saturn, Uranus and Neptune, have both rings and moons. While planetary rings are not see-through, they aren't solid either: they are made up of bits of rocks and ice. The University of Colorado at Boulder's website suggests that these particles are continually recycled through the system because most of them are too bright to have collected much grime. In fact, Saturn pulls material for its rings from one of its moons, Enceladus, which was observed by scientists through the Cassini spacecraft.