The normal force is a force exerted by a flat surface on an object that is in contact with that flat surface.
Note 1: The agent of the normal force is the surface. The surface does not actually have to be flat, but, in most introductory physics problems, it is. The adjective "flat" is used to emphasize the fact that in cases where the object is not in contact with a surface, there is no normal force. For instance, in the case of an object suspended by a string, there is no normal force on the object.
Note 2: The surface in question must be a solid surface rather than the surface of a fluid. The force exerted upon a boat by the water upon which it is floating is called a buoyant force, not a normal force.
Note 3: The normal force is perpendicular to the surface and away from the surface. In fact it is called a normal force because normal means perpendicular.