A Moon is a celestial body that orbits round a planet which in turn orbits round a star normally unless it is a Rogue planet which free-wanders in space. A planet doesn't have to have a moon to be called a planet, after all both Mercury and Venus have no moons orbiting them. From Earth onwards outwards, each planet and dwarf planets (Eris, Pluto) have at least one moon orbiting it. Whilst some moons have names like Miranda and Oberon, our moon is simply known as The Moon.
The first Moon other than the moon to be discovered was discovered by Galileo Galilei who on 7th January 1610 discovered four moons of Jupiter when he looked at the planet through a telescope that he had invented. (Io, Ganymede, Callisto, and Europa) ref: Wiki.
Moons are not just limited to planets in this Solar System, they have been found orbiting Extrasolar Planets (Exoplanets), they are know as Exomoons. According to the University of Washington, life could exist on Exomoons as well. Ref: McMaster University Researchers Therefore the Forest Moon of Moon of Endor and Pandora from Avatar could be grounded in reality.
The moon of Europa has a tenuous atmosphere of Oxygen and has a frozen landscape which it is believed to have a water under the crust. Future missions to the system will try to find out what is under the crust and if there is indeed water and possibly life.
The Moon of Titan is believed to have an atmosphere much like in the same way we have on Earth and on Mars. The atmosphere is not one that we can sadly breathe in but it is the closest of the moons to have an atmosphere like ours.
The Largest Moon in proportion to the size of the planet that it orbits is the The Moon, the one that orbits our planet. However if you are talking about size generally then the largest moon in The Solar System is Ganymede which if it was orbiting The Sun would be classified as a planet. Ganymede is bigger than both Mercury and the furthest (dwarf) planet Pluto. To put it into contents, Ganymede is still only about three quarters the size of Mars.