Before defining 'normal force', it is helpful to define 'normal' in the particular sense that physicists are talking about
this time (
physicists are TERRIBLE about the
term 'normal', we use it for
everything under the sun... although it always
boils down to the same thing). Normal in this case refers to a
vector being
orthogonal (aka
perpendicular) to a
surface. In three dimensions, this
constrains the vector to a
single dimension.
In other words, it determines the
direction of the vector.
Senses of the term 'normal' that this does not refer to include the idea that the vector has a
magnitude of 1 and has no units. I will call this 'completely normal', since it does come up.
Given a force acting on a surface, the normal force is the vector component of the force which acts in the direction of the vector which is normal to the surface. The way one finds the magnitude of this normal force is to multiply via the dot product the vector of the force acting on the surface with the completely normal vector of the surface. So, a glancing blow might have a great deal of force but little normal force. On the other hand, a pile driver is intended to deliver all of its force as a normal force.
The normal force is used for several things. Primarily, it is used to determine the maximum static friction force or the kinetic friction force acting on an object. Both are proportional to the normal force.