Moon

The Moon of Earth II is like Earth's moon Luna. Its significance is that it has caused some Saiyans to transform into their Great Ape form and has been destroyed two times and revived two times.

Prior to Dragon Ball
Goku is known to have transformed into a Great Ape and killed Grandpa Gohan, of which he has no recollection.

Dragon Ball
In Dragon Ball, Goku is shown to place the Monster Carrot and his men on the moon after defeating them using his Power Pole.

Pilaf's Castle
Goku transformed upon seeing the moon when trapped in Pilaf's Castle. Puar had cut his tail so that Goku would return to normal.

21st World Tournament
In the 21st World Martial Arts Tournament, Jackie Chun (Master Roshi) destroyed the moon with a MAX Power Kamehameha so that Goku would revert to his normal state after his transformation.

22nd World Tournament
Man-Wolf, a fighter in the 22nd World Martial Arts Tournament, could not turn back to a human because Roshi destroyed the moon. Master Roshi used a Hypnosis Technique on Man-Wolf to convince him that Krillin's head is the moon. The Hypnosis Technique worked and Man-Wolf turned back to a human.

Moon Revived
It was not mentioned, but the moon must have been revived during Goku's training at Kami's Lookout, as Goku in the 23rd Tournament mentions that his tail was removed so that the moon could be restored. The moon is also seen later, indicating that it has been restored.

Gohan's Metamorphosis
Gohan first transformed into a Great Ape on seeing the moon during his training with Piccolo. To prevent it, Piccolo destroyed the moon. After this, Earth II's moon was never shown again.

Moon's Image
Goku's Space pod projected the image of a moon which again made Gohan transform into a Great Ape. When Piccolo destroyed the Space pod the projector was destroyed.

Fake moon during the battle against the Saiyans
This happened when Vegeta made a Power Ball that transformed him into a Great Ape. Yajirobe cut off his tail so Vegeta could not transform anymore. Later, Gohan transformed after looking at the Power Ball. Vegeta cuts Gohan's tail with a Destructo Disk after a short struggle, but Gohan landed on him and Vegeta retreated.

Wrath of the Dragon
The moon appears in the sky in one scene of the movie Dragon Ball Z: Wrath of the Dragon. It is possible that Dende, after becoming Guardian of Earth, restored the moon when it was clear that there would no longer be threats from evil Saiyans/Great Apes.

Tuffle-Human War
During Dragon Ball GT, Goku was fighting for the humans of Earth in a death-lock battle with the Tuffle King's descendant, Baby, on New Planet Plant. Although, the moon was not present, Goku was able to transform into a Golden Great Ape due the reflection of Blutz Waves by the Earth. Although it was the Earth that caused the transformation, all of the people present for the battle questioned where the moon was, including a brainwashed Bulma and Baby himself, giving an elusion to the moon whereabouts.

Moons of other Planets
Most moons or satellites of other planets have been shown when the planet was shown, and without much significance. However, the moons of Planet Vegeta and of Planet Kanassa were important, as Saiyans had used them to transform. The moon played a significant role during the Saiyan-Tuffle war. It allowed the Saiyans to transform into Great Apes and gave them the victory over the Tuffles after a 10 years war on Planet Plant. New Planet Plant has two moons after being created with the planet by the Earth Dragon Balls.

The sky of Arlia holds two moons. During the destruction of the planet, it shows one moon being destroyed with the planet.

Appearances in Video Games
The moon is a playable stage in Dragon Ball: Shenron no Nazo. Monster Carrot is the boss of this level, and enemies such the four-armed alien named Kurilien are there.

The Tail&Moon card allows Saiyan characters such as Gohan to transform into Great Ape in the Gokuden games.

In Dragon Ball Z: Budokai Tenkaichi 3, the moon is seen in battle stages at night, allowing Saiyans with tail to transform into Great Apes. The moon can also be seen in the computer game Dragon Ball Online.

In Dragon Ball Heroes, it is possible to create a moon using special cards in order to transform into Great Apes or Golden Great Apes. The moon is shown in the sky when Gohan turns into a Great Ape in the story mode of Dragon Ball Heroes: Ultimate Mission.

Trivia

 * The moon is also destroyed in Akira Toriyama's previous manga series, Dr. Slump. Arale Norimaki breaks it by throwing a rock at it in chapter 23. It is shown repaired in chapter 27.