A star is a very specific type of object, a massive body of hydrogen and helium gas that has become so massive that it has started to undergo nuclear fusion in its core.
They are usually by far the most massive object in their gravitational vicinity and dominate their systems, but not always. They emit a tremendous amount of energy from this core nuclear reaction.
The moon is considered a satellite because another larger object (the Earth) gravitationally dominates the orbit.
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