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.