The normal force will be equal to the weight (which is a force) if the object is "resting" on the surface which is producing the normal force. Essentially weight is always apparent because the weight of a 10kg mass on earth is different than its weight on mars which is also different than its weight in an elevator on earth (unless you accelerated the elevator appropriately to match the force the mass "feels" resting on the surface of mars.)
The normal force is the force that is "needed" to prevent the object (or a human in an elevator) from falling through the surface that it is resting on. It is a force that opposes the force that is pressing an object onto a surface. On earth the ground provides a normal force to prevent you from falling through into the center of the earth. A thin piece of paper may be sufficient to provide a very good normal force for an ant (tiny little massed creature) but probably not for a human (on earth of course).
**On a side note, if you were on the sun the force of gravity (aside from the heat melting the paper) would be extremely large and may give the ant enough weigh to rip through the paper (in this case the normal force would fail to do its job which is to apply enough force back to an object to keep it on its surface and not allow it to fall through its surface. Understanding this is key to understanding the normal force.
The scale is exerting a normal force on you. 393
The scale is providing a normal force which counters any force you are imposing on it and thats why the reading changes when the acceleration of the elevator changes (because when the acceleration of the elevator changes your weight changes (because ma=F and when the downward force increases (when elevator is going up) the normal force increases. Or when the downward force decreases (when the elevator is going down) the normal force decreases. The normal force is equal to the weight. So if you know the normal force you know the weight and vise versa. They are equal but opposite forces. Hope this helps. 797