the tangent plane
Stand on a smooth hillside and look at the patch of ground right under your feet. Over a small enough area it looks flat, even though the whole hill curves. The flat plane that best matches the surface at the point where you stand — the plane the surface would lie in if you only looked at an infinitesimal neighbourhood — is the tangent plane. It is the two-dimensional analogue of the tangent line to a curve: the surface's best flat approximation at one point.
Precisely, fix a point p on a regular surface S and a parametrization x(u, v) with x(u_0, v_0) = p. The two partial-derivative vectors x_u and x_v at that point are tangent to the surface (they are velocities of the coordinate curves through p). Because the surface is regular, x_u and x_v are independent, so they span a genuine plane through p — that plane is the tangent plane, written T_p S. Every tangent vector to the surface at p — the velocity of ANY smooth curve on S passing through p — is a combination a*x_u + b*x_v of these two basis vectors. So T_p S collects all the directions in which you can move while staying (to first order) on the surface. The single direction perpendicular to this whole plane is the surface normal.
The tangent plane is the stage on which surface calculus is performed. The first fundamental form lives on it (it is just the dot product restricted to tangent vectors), the Gauss map and second fundamental form are read off by watching how the normal tilts as you move within it, and a geodesic is a curve whose acceleration has no component inside the tangent plane. One caution: T_p S is an honest plane through p with a definite position in space, but for many purposes we slide it to the origin and treat it as a two-dimensional vector space of directions — the 'tangent space.' Both pictures are standard; do not be surprised to see the tangent plane drawn touching the surface, or drawn as an abstract plane of arrows.
At the top point (0, 0, 1) of the unit sphere x^2 + y^2 + z^2 = 1, the tangent plane is the horizontal plane z = 1: it touches the sphere at exactly that one point and lies flat against it. The basis vectors x_u, x_v there both lie in this horizontal plane, and the surface normal points straight up along (0, 0, 1). Any direction you could 'roll a tiny ball' at the top of the sphere lives in this plane.
The tangent plane touches the sphere at one point; the normal points outward there.
The tangent plane touches the surface but generally does NOT stay near it away from the contact point — at a saddle point the surface pokes through its own tangent plane on two sides. 'Tangent' means agreement to first order at one point, nothing about the surface lying on one side of the plane.