Earth's moon does have a name: In English, it's "the moon." The word moon is Proto-Germanic in origin, deriving from a similar-sounding word that came into use a few thousand years ago in Northern Europe.
For most of human history, there didn't need to be a more specific term to differentiate our moon from other moons that orbit other planets in the solar system, and for good reason: we didn't know there were any other moons. "Until Galileo discovered that Jupiter had moons in 1610, people thought that the moon was the only moon that existed," NASA's lunar science website states.
"After other moons were discovered," the NASA site continues, "they were given different names so that people would not confuse them with each other. We call them moons because they orbit planets the same way that the moon orbits around Earth." [Read: How the Moon Got There ]
The four biggest of Jupiter's 64 moons — the ones discovered by Galileo in 1610 — are named Io, Europa, Ganymede and Callisto.
In Latin, our satellite's name is "Luna." Because a significant chunk of English comes from Latin, many terms associated with the moon are related to this Latin name — for example, the adjective "lunar," and the noun "lunatic," an old-fashioned word for a mentally ill person. (Madness was thought to be correlated with the phases of the moon.)
In Greek, our moon is named "Selene," as is the moon goddess of ancient Greek mythology. The English word "selenology," or the study of the moon's geology, derives from it.