Waves in a Large Free Sphere of Water - An experiment at the International Space Station. What humans perceive as "weight" is not actually the force of gravity pulling us towards the ground (actually, towards the center of the Earth — although this is the technical definition of "weight"). What we feel as "weight", is actually the normal reaction force of the ground (or whatever surface we are supported by) "pushing" upwards against us to counteract gravity's downward pull — that is: the "apparent weight". (In the remainder of this article, the term 'weight', without 'apparent', is used in this sense.) While this is not always intuitive, imagine the floor dropping out from under you: without it, you'd be falling — and experiencing weightlessness. It's the floor supporting you against gravity's pull — and which keeps you from falling to the center of the Earth — that creates the sensation of "weight".
For example: a person in a broken lift in free-fall "experiences" weightlessness. This is because there is no force from the lift's floor on the person's feet, against the pull of gravity, as both the lift and the person are being pulled down with the same acceleration. When the lift is at rest on the ground, however, the force of gravity pulling downwards on the person is exactly matched (in the opposite direction, and by the same amount) by the support of the lift floor.
Because the person's skeleton is solid, each horizontal cross section of the person experiences not only the force due to gravity on it, but also the weight of whatever portion of the person is above it. (In the case of an object, or portion thereof, which is not supported from below, but suspended from above, a 'negative pressure', or tension gradient exists. It occurs because each cross section of a hanging object, a rope for instance, must support the weight of every piece below it.) Part of feeling "weight", then, is actually experiencing such a pressure/tension gradient within one's own body parts (e.g.: while standing on one foot, the foot on the ground would feel the pressure of the entire body's weight, whereas the other leg and both arms would feel/be subjected to the tension gradients of their own weight being pulled down against their sockets).
In free-fall, a person or object experiences no measurable (or apparent) weight because all parts of the object are accelerating uniformly (any variations in acceleration due to tidal forces being imperceptible).