A player can use Deathstars to attempt to destroy another player's moon. When a moon is destroyed, it does not leave behind any debris field. The deathstar is the only ship that can perform this mission.
The chance that the moon is destroyed is based on the number of Deathstars sent on the destroy mission and the size of the moon. See the article on Moons for more details.
Players destroy moons for a number of reasons, the most common of which is that the player who owns the moon (the victim) represents a threat to the attacking player's fleet or to other players in his/her alliance. Destroying the moon will prevent the victim from performing Sensor Phalanx scans on the surrounding area, eliminating the threat to the attacking players or alliance. It will also discourage the victim from further interfering with the player or alliance that attacked him/her, as moons are somewhat difficult to obtain and develop. Moons allow you to fleetsave without having your fleet phalanxed, so moon destroying can be a good strategy if you want to make your enemy's fleet more vulnerable.
Popping a moon is an effective way to make a fleet vulnerable. If someone does a recycle fleet save, attack, or transport and the moon is destroyed while the fleet is in flight, the activity will transfer to the planet. Once the fleet activity is transferred an attack can be timed immediately following the fleet's return.
You can calculate the chance of destruction with a moon destruction calculator.