what happens to Gradient Vector at critical points of a multi variable function

Could any one please explain what happens to the gradient vector at critical point....it becomes a zero/null vector ..right??....But a zero vector should nt have a direction but on the other hand what I understood is at every point on the level curve of function the gradient is orthogonal to the level curve.......Then what happens at the critical point (its just a Null vector , so it simply gets vanished at those points and should nt have any direction)..... This is even more ambiguous if you consider the lagrange multiplier method ..where both the the objective function and constraint function share the same tangent line/ tangent plane and both have the gradient vector perpendicular to their respective function level curves at the crtical points .... thats what a video lecture from MIT says....AM wrong in understanding may be...but would love to get clarified.... So pls reply