Answer:
Mars has two natural satellites, Phobos and Deimos. There are also 3 functioning artificial satellites there: Mars Reconnaissance Orbiter, Mars Odyssey, and Mars Express.
Artificial satellites that are no longer functioning, but still in orbit, around Mars include Mariner 9, the Viking 1 orbiter, and probably the Mars Global Surveyor, though after the MGS entered safe mode the MRO team unsuccessfully tried to image it to determine its orientation, so it may no longer be in orbit; its apapsis is close enough to the surface for atmospheric drag to be a potential factor.
Mars 2, Mars 3, and the Viking 2 orbiter may still be in orbit as well; I couldn't find any hard data one way or the other.