In physics, weight is a measure of the force exerted by gravity on a mass.
You probably know that you weigh less on the Moon than on Earth. For instance, if you weigh 100. pounds on Earth, you will weigh 16.6 pounds on the Moon. But, if your mass on Earth is 100 kg, your mass on the Moon is... also 100 kg. Because the amount of matter you have does not change from the Earth to the Moon, but the gravitational force on the Earth is stronger than on the Moon, so you weigh more on Earth.
You can think of gravity pulling a mass toward the center of an object like the Earth. It pulls a lot harder for more massive objects like the Earth than for the Moon. That's why there's a difference in weight.
As a caveat, adding energy or mass to an object will affect its mass. Additionally, general relativity informs us that when something as traveling very near the speed of light, the whole idea of mass equivalency is not exactly true...