second fundamental form
The first fundamental form is what the ant on a surface can measure from inside; the second fundamental form is what a bird flying above can see that the ant cannot — namely how the surface curves away from its own tangent plane into the surrounding space. Stand at a point on a surface, lay down the flat tangent plane that just touches it there, and watch how the surface peels off that plane as you move in different directions. That peeling is exactly what the second fundamental form records.
Precisely, let n be the unit normal to the surface at a point. As you take a small step (du, dv) the surface lifts off its tangent plane by an amount whose leading behavior is the quadratic form II = L du^2 + 2M du dv + N dv^2, where L = r_uu dot n, M = r_uv dot n, and N = r_vv dot n (the components measure how the second derivatives of the surface point along the normal). Feeding a tangent direction into II measures the normal curvature in that direction — how sharply the surface bends, in space, as you head that way. The principal curvatures kappa_1 and kappa_2 are the largest and smallest such normal curvatures, achieved in the perpendicular principal directions; they are the eigenvalues of the shape operator, the linear map built from II together with I. From them follow the two famous curvatures of a surface.
The second fundamental form is the carrier of extrinsic curvature — the part of a surface's shape that depends on how it is embedded, not just on its internal metric. It distinguishes a flat sheet (II = 0, no bending into space) from a rolled cylinder (which bends one way, II nonzero) even though the two share the same first fundamental form. It governs how light reflects off a curved mirror, how a thin shell or membrane resists bending, and how a soap film balances surface tension (minimal surfaces are exactly those with zero mean curvature, a statement about II). The deep counterpoint, Gauss's Theorema Egregium, is that although L, M, N are extrinsic, the particular combination giving the Gaussian curvature turns out to be intrinsic — bending alone, without stretching, cannot change it.
On a saddle surface like z = x^2 - y^2, the surface curves upward as you walk one way and downward as you walk the perpendicular way. The second fundamental form has opposite signs in those two principal directions, so the principal curvatures kappa_1 and kappa_2 have opposite signs — the hallmark of a saddle (negative Gaussian curvature). On a sphere both principal curvatures share a sign: it curves the same way everywhere.
Opposite-sign principal curvatures mark a saddle; same-sign mark a dome. II reads the bending into space.
The second fundamental form is extrinsic: it depends on the chosen direction of the unit normal (flipping n flips its sign) and on how the surface sits in space, unlike the intrinsic first form. Its sign conventions vary between textbooks, so always check which normal orientation an author uses.