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.
To add to what is above, the normal force is the force that prevents objects from simply passing through each other. It is the electromagnetic repulsion of the negatively charged valence electrons in the atoms of the objects in quesiton.