Mars only has two moons. Although this is more than us here on Earth, it isn't very many compared to some of the other planets in the solar system.
The moon's names are Phobos and Deimos.
Both of the moons are thought to be asteroids that flew too close to the planet at some point, and were 'captured' by the planet's gravity. They now orbit Mars continuously.
Like a lot of moons, they are named after characters in Greek mythology. I think the story behind their names are quite interesting. Phobos and Deimos were brothers, and their father was a character named Ares, who was the God of War. The two brothers followed there father into war. Ares was then know to the Romans as 'Mars'.
I like this story as now it seems like the planet Mars is still father to the two smaller moons.
Both moons were discovered on the same day, 12th August 1877, by Asaph Hall.
From the surface of Mars, the moons would look very small. Phobos would look about a third as big as our moon does to us. Deimos would look like nothing more than a very large star. In comparison to this, the sun would look huge from Mars. It is safe to say that there would be no lunar eclipses on Mars, the moons being far too small to create the phenomenon.
This image shows the comparison of Earth's moon next to Mars' moons, as viewed from their planets.