It can be said to have no weight, as its weight could only be expressed if it were stationary in the Earth's gravitational field. However its mass (the matter it contains) is about 7.35x1022 kg. This is about 1/80th the mass of the Earth.
There is a large distinction between weight and mass. Mass has to do with the amount of matter is in some object, and determines how hard it is to push something around. Weight is how big the force of gravity is that is acting on some object. An object has the same mass no-matter where it is, but its weight depends on being close to the Earth or some other planet. For example, if you were out in space, far away from any planets, you would have no weight since there is no gravity, but your mass would still be the same as it is here on Earth.
So, it makes more sense to talk about the mass of the moon than its weight.
Relative Weight of Objects
It is usually not appropriate to talk about the weight of astronomical objects because the force of gravity varies, depending on the mass of a pair of objects, and their distance from one another. For example, if Earth attracts a table with a force of 100N, then the table attracts the Earth with the same force of 100N (Newton's Third Law), so the weight of Earth - on that table - is 100N. If you took the table millions of miles from Earth, that weight (the table or the Earth) would be much smaller, because the effect of Earth's gravity would be much less.
Instead of asking about the weight of objects, compare the objects' masses - those don't change, or hardly change.
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