The "official" name is the Moon, with a capital M. The satellites of the other planets are called moons (small m). Similarly, stars are referred to as suns, but our star is the Sun.
You may occasionally - especially in old poems - see the moon called 'Diana' or 'Luna'...these being names associated with the myths and language of the Romans. (Our word 'lunatic' comes from the idea that mad people are somehow influenced by the moon.)