The weight force that we actually sense is not the downward force of gravity, but the normal (upward) force exerted by the surface we stand on, which opposes gravity and prevents us falling to the center of the Earth. This normal force, called the apparent weight, is the one that is measured by a spring scale.
For a body supported in a stationary position, the normal force exactly balances the earth's gravitational force, and so apparent weight has the same magnitude as actual weight. (Technically, things are slightly more complicated. For example, due to the earth's rotation objects are subject to a small centrifugal force, varying with latitude, which partially offsets gravity. The normal force therefore balances a force slightly less than the true force of gravity. These and other factors are explained further under Apparent weight.)
If there is no contact with any surface to provide such an opposing force then there is no sensation of weight (no apparent weight). This happens in free-fall, as experienced by sky-divers and astronauts in orbit, who feel "weightless" even though their bodies are still subject to the force of gravity. The experience of having no apparent weight is also known as microgravity.