The weight of a thing is the force with which the Earth pulls it downward. Weight is what your arm fights when you hold a bucket, and what stretches the spring when the bucket hangs from it. Like every force, weight has a direction: straight down, always.
Examples
Example 24.3 (Cousins, not twins)
Mass answers: how much stuff? It is a property of the thing alone — the same in your hand, underwater, or on the Moon. Weight answers: how hard is it pulled down here? It depends on the thing and on where it is. Change nothing but the place, and the mass stays; the weight can change.
Example 24.6 (The Moon is a weaker puller)
The Moon is much smaller than the Earth, and it pulls things far more weakly — about six times more weakly. An astronaut’s pack keeps every gram of its mass on the trip, but its weight up there shrinks to a sixth. That is the whole secret of the bouncing: legs trained to fight Earth-pull suddenly fight a gentle Moon-pull, and every step becomes a leap.