The Moon is definitely not a star. Stars have to be very massive - about 15 times the size of Jupiter or larger - in order to burn hydrogen, which is the main definition of a star. The Moon doesn't burn hydrogen, and it doesn't produce it's own light - it's just reflected light from the Sun.

The Moon isn't a planet either. It doesn't orbit just the Sun, like the Earth or Jupiter - it goes around the Earth first, then the Sun. So it's technically a satellite.