the normal force

The normal force is the push a surface gives back to whatever presses on it. Set a book on a table and the table pushes up on the book; lean on a wall and the wall pushes back on you. It is why solid surfaces feel solid. 'Normal' here is a geometry word meaning perpendicular.

The normal force is the component of a contact force that is perpendicular (normal) to the surface, directed away from the surface. It adjusts itself to whatever is needed to prevent objects from passing through each other. On a flat horizontal table with nothing else pushing, it balances gravity, so N = m g. On a surface tilted at angle theta, only the component of weight perpendicular to the surface must be balanced, so N = m g cos(theta), smaller than on the flat.

The normal force is not always equal to the weight. Press down on the book and N grows; pull up on it and N shrinks; on a slope or in an accelerating elevator, N differs from m g. It is a response force whose size is set by the requirement of no penetration, found from Newton's second law along the perpendicular direction.