In everyday talk, we say things fall because the Earth's gravity pulls on them. We talk as if our weight was a "given". Actually, weight changes when the pull of gravity changes. The Moon is much smaller and the pull of gravity on the Moon is about 1/6th that of Earth. So any object on the Moon weighs 1/6th of its weight on Earth. What does not change is the amount of matter in an object. That is called conservation of mass. On Earth, mass and weight are the same for most purposes, though a sensitive gravimeter can detect the difference. The difference can be very different on another world such as the Moon.