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Your weight is the force that is exerted on you by gravity. Your weight is what pulls you down to earth, or indeed any planet or moon that you happen to be on.
Your mass is simply a measure of how much matter there is. It is responsible for the force of gravity and so the greater the mass of either you or the celestial object that you're standing, then the gravitational force acting between you and the object will increase, and therefore so will your weight.
Hence, as the local gravitational field strength is weaker on the moon compared to on earth (because of its lesser mass), on the moon your weight is also lesser, but your mass will remain the same.
In everyday usage, measuring weight is a method of finding your mass. By measuring your weight, we can factor out the gravitational field strength, which is roughly constant across the planet, and find your mass. Thus we tend to talk about weighing ourselves, even though kilograms and stones are a measure of mass.
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