Weight = mass x gravitational field strength
Gravitational field strength is determined by two factors: how large the planet/moon/body is, and how close the object is to that body. Discounting the second factor (which has a minute effect on field strength over distances below 100km or so), the size of the planet/moon/body is proportional to the field strength.
On Earth, the field strength is roughly 9.81N/kg (where kg is the SI unit of mass and Newtons are the SI unit of weight) - this means that every kilo of mass weighs just less than 10N here.
Conversely on the moon, the field strength is roughly 1.6N/kg, meaning that every kilo weighs only 1.6N there. The field strength is so much smaller because the moon has less that 1/81th of the Earth's mass.
In conclusion, take the example of a a 5kg bowling ball: on Earth it would weigh 5 * 9.81 = 53.955N whereas on the moon it would weigh 5 * 1.6 = 8kg. They have the same mass in both places, but weigh very different amounts.
