Tangent Plane

A Plane that is tangent to a Surface.

Level Surface

When the surface in question $S$ is given by $F(x,y,z)=k$, and there is a point on the surface $P(x_0,y_0,z_0)$ $\nabla F(P)$ is normal to the surface at the point $P$ Therefore, we have been supplied a normal vector and a point, so we have enough information to create a plane.