a line of curvature
At every point of a surface (away from the special umbilic points) there are two perpendicular directions in which the surface bends most and least — the principal directions. Now imagine starting at a point and always walking in the 'bends most' direction, letting it gently steer you as it changes from point to point. The smooth path you trace is a line of curvature. There are two families of them, woven into a perpendicular grid laid naturally across the surface, like the seams that the surface's own bending wants to follow.
Precisely, a curve on the surface is a line of curvature if at every one of its points its tangent direction is a principal direction — that is, it always heads along a direction where the normal curvature is a maximum or a minimum. Equivalently (Rodrigues's criterion), along such a curve the change in the unit normal N is parallel to the change in position: dN = -k dx for a scalar k, which is then the principal curvature in that direction. Through each non-umbilic point pass exactly two lines of curvature, one for each principal direction, and they cross at a right angle. Choosing coordinates (u, v) so the coordinate curves ARE the lines of curvature makes the formulas wonderfully simple: both F = 0 (in the first fundamental form) and f = 0 (in the second) at once, and the principal curvatures become just e/E and g/G.
Lines of curvature matter both in theory and in practice. They give the cleanest coordinate system for computing curvature, and they appear physically: on a smooth surface they are the directions along which a rolling cylinder would not wobble, and in optics the lines of curvature of a lens surface organise astigmatism. A caution: lines of curvature are NOT the same as geodesics. A geodesic is a locally shortest path (zero geodesic curvature); a line of curvature follows the principal bending directions. They sometimes coincide — for instance the meridians of a surface of revolution are both — but in general they are different curves answering different questions.
On a surface of revolution (think of a vase made on a potter's wheel), the lines of curvature are exactly the meridians (the vertical profile curves, running top to bottom) and the parallels (the horizontal circles at each height). They cross at right angles everywhere and are the most natural grid for the surface. On a torus the same is true: the circles around the tube and the circles around the hole are the two families of lines of curvature.
On a surface of revolution the meridians and parallels are the lines of curvature.
Lines of curvature break down at umbilic points, where every direction is principal so there is no distinguished 'most-bent' direction; the two families of lines of curvature can swirl into singular patterns around such points, much as the meridians of a globe all converge at the poles.