the principal curvatures
Stand at a point on a surface and slowly turn on the spot, each time asking how sharply the surface curves in the direction you face. As you rotate, the normal curvature rises and falls. There will be one direction where the surface curves the MOST and another where it curves the LEAST. Those two extreme amounts of bending are the principal curvatures, k_1 (the largest) and k_2 (the smallest); the directions in which they occur are the principal directions.
Here is the precise structure. The normal curvature in direction w is k_n(w) = II(w, w), and as w runs over all unit tangent directions this is a smooth function on a circle, so it has a maximum k_1 and a minimum k_2. A clean theorem (from linear algebra applied to the shape operator dN) says k_1 and k_2 are the eigenvalues of the shape operator, and their principal directions are its eigenvectors — which are always perpendicular to each other when k_1 is not equal to k_2. Euler's formula then reconstructs every normal curvature from just these two: k_n = k_1 cos^2 theta + k_2 sin^2 theta, where theta is the angle from the first principal direction. So the entire bending of the surface in all directions is encoded by two numbers and two perpendicular axes.
The principal curvatures are the raw material from which the two famous curvatures are built: the Gaussian curvature is their product K = k_1*k_2, and the mean curvature is their average H = (k_1 + k_2)/2. Their signs tell the local shape at a glance: both positive or both negative means a dome or bowl (elliptic point); opposite signs means a saddle (hyperbolic point); one of them zero means a trough that is flat in one direction (parabolic point). A caution on signs: individually k_1 and k_2 depend on the chosen direction of the normal — flip the normal and both change sign — so the mean curvature's sign is a convention, but their PRODUCT K is sign-independent and so is genuinely intrinsic.
At a point on a torus (doughnut surface), the principal directions are 'around the tube' and 'around the hole.' On the outer equator, both principal curvatures are positive (the surface domes outward), so K = k_1*k_2 > 0: an elliptic point. On the inner equator (the throat nearest the hole), one principal curvature is positive and one negative — the surface curves toward you around the tube but away from you around the hole — so K < 0: a hyperbolic (saddle) point. On the top and bottom circles, one principal curvature is zero, giving parabolic points with K = 0.
On a torus the same surface has elliptic, hyperbolic, and parabolic points.
When k_1 = k_2 (an umbilic point, e.g. every point of a sphere) the surface curves equally in all directions, the maximum and minimum coincide, and the principal DIRECTIONS become undefined — there is no special axis, just as a perfectly round bowl has no preferred way to tilt.