Let P be a point on a smooth surface. The tangent at P to any curve through P on the surface is called a tangent line at P, and the tangent lines are all perpendicular to a line through P called the normal to the surface at P. The tangent lines all lie in the plane through P perpendicular to this normal, and this plane is called the tangent plane at P. (A precise definition of ‘smooth’ cannot be given here. A tangent plane does not exist, however, at a point on the edge of a cube or at the vertex of a (double) cone, for example.)
At a point P on the surface, it may be that all the points near P (apart from P itself) lie on one side of the tangent plane at P. On the other hand, this may not be so—there may be some points close to P on one side of the tangent plane and some on the other. In this case, the tangent plane cuts the surface in two curves that intersect at P. This is what happens at a saddle-point.