normal curvature
Stand at one point on a hilly surface and ask: 'how sharply does the ground curve if I walk THIS way?' Walk one way and the ground might rise gently; turn ninety degrees and it might fall away steeply. The curvature you feel depends on the direction you choose. The normal curvature is the number that answers 'how much does the surface bend in this particular direction,' measured purely by the bending toward or away from the surface's normal.
Precisely, fix a point p, a unit tangent direction w, and slice the surface with the plane containing w and the unit normal N. That plane cuts the surface in a curve (a 'normal section'). The normal curvature k_n(w) is the signed curvature of this slice curve at p — positive if it bends toward N, negative if away. There is a clean formula: k_n(w) = II(w, w), the second fundamental form evaluated on the unit direction w. Another route (Meusnier's theorem) says that for ANY curve on the surface through p with unit tangent w, the component of its curvature vector along the normal equals k_n(w); only the IN-surface (geodesic) part of the curvature depends on the particular curve. So normal curvature is a property of the direction, not of which curve you happened to draw in that direction.
Normal curvature is the bridge from a single number per direction to the whole curvature picture. As w sweeps once around the tangent plane, k_n(w) varies, and Euler's formula gives it as k_n = k_1 cos^2 theta + k_2 sin^2 theta, where k_1, k_2 are the principal curvatures (the maximum and minimum of k_n) and theta is the angle from the first principal direction. So every normal curvature lies between k_1 and k_2. One honest caution: the SIGN of the normal curvature depends on the chosen direction of the normal N. Flip N and every normal curvature flips sign; what is genuinely meaningful without that choice is the PRODUCT k_1*k_2 (the Gaussian curvature), since a double sign-flip cancels.
On a cylinder of radius R, walk along the axis direction: that slice is a straight line, curvature 0, so the normal curvature there is 0. Now walk around the circular direction: that slice is a circle of radius R, so its curvature is 1/R, and the normal curvature is 1/R (toward the axis). Every other direction gives a value between 0 and 1/R by Euler's formula. These two extremes, 0 and 1/R, are exactly the principal curvatures of the cylinder, and their product 0 * (1/R) = 0 is its Gaussian curvature — the cylinder is intrinsically flat.
On a cylinder the normal curvature ranges from 0 (along the axis) to 1/R (around it).
Normal curvature only captures the part of a curve's bending that points along the surface normal; the part lying within the tangent plane is the geodesic curvature, a separate quantity. A curve can be sharply curved as a space curve yet have small normal curvature if most of its bending is in-surface.