If an object weighs 6 N on earth what will be its weight on moon? What will be the change in its mass?

The weight of body on moon will be 1 N i.e. [6N x $\dfrac{\text{1}}{\text{6}}$] because the weight of body is $\dfrac{\text{1}}{\text{6}}$ times lighter on the moon than it is on earth.

There will be no change in its mass because mass of a body remains constant irrespective of its place.