Great question! First it is important to define weight. Weight is the amount the planet is pulling a mass down. The difference between the pull of earth's gravity and the pull of the moon's gravity is 1/6. You are correct that one would weigh 1/6 of what you weigh on earth than what you weight on the moon.

It is important to not confuse weight with mass. One's mass does not change where ever one is located. If you are 50 kg of mass on earth you are also 50 kg of mass on the moon. Mass is a measure of how much stuff is there. Weight is a measure of how much stuff is being pulled down.