What is the Normal Force? This is a force of CONTACT. If there is no contact between the object of interest and some other surface, e.g. a floor, there is NO Normal Force.

Consider the picture below which shows a box sitting on a floor.


If the box is indeed sitting on the floor and is not moving, then we know that there must not be a Net Force acting on the ball, otherwise it would start moving.

If we now want to figure out what individual forces are acting on the object, we can immediately say that the object has weight. This is a force.
But, since the box does not move through the floor, something must be holding it up !! That is the floor. In order to do this, the floor must exert a force on the box to counteract the weight (as there are no other forces present in this case). Hence, a Free Body Diagram for the ball would look like the following:

We call the force that the floor exerts on the box a Normal Force as the force acts NORMAL to the surface upon which the box is resting. Note, Normal forces can be between ANY two objects as long as the force acts perpendicular to the surface of contact... (check out this example)


To review: A NORMAL FORCE