The force that acts on the mass of an object because of gravity is called its weight. Therefore weight is a force and is measured in newtons.
Weight is the force of gravity acting on a mass that is positioned in a gravitational field. This changes with the strength of the gravitational field.
There is an equation that links mass to weight in a gravitational field:
On Earth g = 10N/kg. Therefore my weight is 650 Newtons.
This is only true whilst I stay on Earth. On the moon, I would be very light and jumping would be very easy. This is because g (Moon) is only 1.67 N/kg. I would weigh a sixth of what I did on Earth, but it wouldn't mean I was any thinner.. I'd still have all those atoms and molecules that make up my mass!