A normal force is one that is 'normal' or perpendicular to the plane of action, as opposed to a shear force which acts parallel to the plane of interest.
For an object (e.g., block) on an a 'flat surface', the normal force points perpendicularly to the flat surface. No motion would occur in the normal direct, but that force would be static and balance by a force pushing against the normal force. If friction is considered, the friction is perpendicular to the normal force, or parallel to the 'flat surface'.
There is a discussion here - hyperphysics.phy-astr.gsu.edu/hbase/frict.html - but the site seems to be down at the moment.