Below is an interactive illustration of a function of x and y. The z axis is the value of the function.
The point, , can be moved with the sliders which change the and values of the point. Dashed lines on the surface show the functions values along constant or values. There are several check boxes that allow you to show various parameters of the function at the point .

Normal Line - shows a line perpendicular to the surface

Tangent Plane - shows a plane that is tangent to the surface. Any line in this plane through will also be tangent to the surface.

Tangent Line (x=x_{A}) - shows a line in a constant plane that is tangent to the surface. The plane is also shown lightly.

Tangent Line (y=y_{A}) - shows a line in a constant plane that is tangent to the surface

Gradient Vector - shows the gradient vector which is in an plane and points in the direction of increasing function value

Contour Line - shows a contour curve where the function value is constant

Moving the view around is a good way to see the three dimensional features. Try viewing the the gradient and contour line from directly above the surface. Notice the angle between the vector and the curve.
Equations for all of the items are shown when the box is checked.