Weight is defined as the measure of the gravitational attraction between two physical objects.
Gravity is defined as the force that attracts two physical objects towards one another. However, the mechanism behind how gravity works is still largely a mystery. Gravity is affected by two variables. The first variable is the mass of the object (the larger the object, the greater the gravity). The second variable is the distance between the two physical objects (the greater the distance between the object, the lesser the gravity).
The moon is only one-fourth the size of the Earth. Thus, because the moon is of a smaller size, the amount of gravity it has on other objects is lower than that which is exerted by Earth. Therefore, you weigh less on the moon.