Explain why the weight of an object is different on earth and on moon even though the objects mass is the same in both place.

The Mass of the Earth is larger than that of the moon resulting in a larger Fg because the strength of gravity on earth is stronger.

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.