The Moon is classed as a satellite – that is, a body that orbits around a planet. Other planets in the solar system have moons, such as Mars, which has Deimos and Phobos. Jupiter has loads, including the largest, Io, Europa, Ganymede and Callisto, which were known to Galileo and can be seen through a small telescope. Saturn has the mysterious Titan which is swathed in a deep atmosphere of organic smog.
A planet is a body that orbits the Sun and must be of a certain size, which is why poor Pluto is no longer classed as a planet, but a ‘dwarf planet’.
The funny thing is, if the Moon were orbiting the Sun like any other planet, in its own orbit, it would be called a planet as it is big enough to fit the criteria! It is larger than Pluto.
The word planet comes from Greek words which mean wandering star, as the first planets looked like stars which moved against the background of stationary stars. But for something to be classed as a planet it needs to orbit a star, and the moon orbits the Earth.