mean curvature
If the Gaussian curvature K answers 'what intrinsic shape is this point — dome, saddle, or flat?', the mean curvature H answers a different, more physical question: 'on average, how does the surface bulge out of its tangent plane?' It is the quantity a soap film responds to, the one that tells you the surface tension's effect, and the one that vanishes precisely for the surfaces that minimise area.
Precisely, the mean curvature at a point is the average of the two principal curvatures: H = (k_1 + k_2)/2. (Some books use the sum k_1 + k_2 instead and call THAT the mean curvature; watch for the factor of 2.) In a parametrization it is H = (e*G - 2f*F + g*E) / (2(E*G - F^2)). Geometrically, averaging k_1 and k_2 sums up the bending across all directions: in fact H equals the average of the normal curvature k_n(theta) taken over all directions theta, a fact that follows from Euler's formula. Because it is an AVERAGE rather than a product, H mixes the two principal curvatures additively — so on a saddle, where k_1 and k_2 have opposite signs, they can cancel and give H = 0 even though the surface is far from flat.
Mean curvature governs the physics of surfaces. The Young-Laplace law says the pressure difference across a soap film or a liquid surface is proportional to H, which is why small bubbles (large H) hold higher internal pressure than large ones. Surfaces with H = 0 everywhere are minimal surfaces: they are the area-minimising shapes a soap film spans across a wire loop, balanced because the bending in one principal direction exactly cancels the other. Two cautions. First, unlike K, the mean curvature H is EXTRINSIC and its sign depends on the chosen normal — flip the normal and H changes sign, so 'H = 0' is meaningful (sign-free) but the value and sign of a nonzero H carry a convention. Second, H = 0 does not mean the surface is flat; it means the principal curvatures are equal and opposite, the hallmark of a minimal surface, not of a plane.
A sphere of radius R has k_1 = k_2 = 1/R, so H = (1/R + 1/R)/2 = 1/R — nonzero, and a smaller bubble has larger H and thus higher internal pressure, matching the Young-Laplace law. A flat plane has H = 0 trivially (both principal curvatures zero). A catenoid (the surface a soap film makes between two rings) has at every point k_1 = -k_2, so H = 0 while K < 0: it is genuinely curved (saddle-like everywhere) yet has zero mean curvature, the signature of a minimal surface.
Sphere H = 1/R; catenoid H = 0 yet K < 0 — a minimal surface, not a flat one.
Mean curvature is extrinsic and sign-dependent (flip the normal and H flips sign), whereas Gaussian curvature is intrinsic and sign-free. So K, not H, survives bending without stretching; do not expect the mean curvature to be preserved when you roll a surface up.