From Dragon Ball Encyclopedia, the ''Dragon Ball'' wiki
The Moon (月, Tsuki) is a celestial satellite that revolves around Planet Earth as in the real world. Its significance is that a full moon (満月, mangetsu) enables Saiyans to transform into their Great Ape forms. Other planets sometimes have "moons".
The Moon was destroyed twice and recreated once.
The Kaioshin Realm had many moons.
Prior to Dragon Ball
It was not mentioned, but the Moon must have been recreated sometime during Goku's training in the Heavenly Realm because he mentions in the 23rd Tenkaichi Tournament that his tail had been removed so that the Moon could be recreated. The Moon was also seen later, thus indicating that it had been recreated.
Dragon Ball Z
Son Gohan first transformed into a Great Ape upon seeing the Moon during his training with Piccolo. Piccolo had to destroy the Moon so that Gohan could turn back to normal. The Moon was never shown again.
Goku's Attack Ball later projected the image of a moon and made Gohan transform into a Great Ape once again. The Attack Ball was destroyed when Piccolo figured out that the image of a moon was being projected from it.
Most moons of other planets were shown whenever the planet was shown, but did not have much significance. The moons of Planet Vegeta and Planet Kanassa, however, were important because the Saiyans used them to transform in battle against their enemies.