Gaussian curvature is one of the most beautiful properties in geometry, because it is a way of describing the curvature of a surface which is dependent only upon the surface’s intrinsic geometry. This is essentially the content of Gauss’ Theorema Egregium (Most Remarkable Theorem) which states that Gaussian Curvature, K, is invariant under local isometry.

What does this mean? One way Gauss liked to think about surface geometry was to consider what a two-dimensional being living on the surface would be able to determine about the surface just by doing measurements of the distance between two points. For example, a person living on the surface of a sphere could determine he was not on a flat plane by measuring the angles of a triangle on the sphere, and noting that the sum of the angles is greater than 180 degrees (in a flat plane, the angles in a triangle always add up to 180 degrees, but try drawing a large triangle on a ball. You can make all three angles 90 degrees if you like!). Essentially, if you had two surfaces which a two-dimensional being could not tell apart just by doing distance and angle measurements, they are said to be isometric. Another way of thinking of isometry is to imagine bending a surface without deforming it, like rolling a flat plane into a cylinder. In contrast, there is no way one could bend a flat piece of paper to cover a sphere, or flatten out the surface of a sphere without deforming the surface. This is essentially the reason why maps of the earth are so distorted, and Greenland often appears to be the size of Africa.

Getting back to the subject of Gaussian Curvature, it is a property intrinsic to the surface, having no dependence on how it’s bent around in three dimensions. A lot can be determined about local geometry by just looking at whether K is positive or negative. Here are some examples which can be realized in three dimensions (note that, although K can vary from point to point, if you’re looking in a small enough region of the surface, any type of surface with K > 0 must "look like" any other type of surface with K > 0 , and the same goes for K < 0 or K = 0):

K > 0: Sphere, ellipsoid, paraboloid, the outer half of a torus (donut).

K = 0: Plane, cylinder, cone, generalized cylinder.

K < 0: Saddle surface (z=xy), catenoid, helicoid, the inner half of a torus (donut).

Essentially, when the curvature is positive, the local geometry looks like a “bump”, and when it’s negative, the local geometry looks like a “saddle”. When K=0, the surface is said to be flat, and the local geometry looks like a flat plane. Wait a minute, you say. “A cylinder looks like a flat plane?” Yes it does, to a two-dimensional being living on the surface of the cylinder, because distances along a cylinder are exactly the same as distances along a flat plane, but “straight lines” are just mapped into helices (spirals). The two-dimensional guy can’t tell the difference, unless he tries walking in one direction for awhile, and ends up back where he started.

If you add up, or integrate the Gaussian curvature at all points on the surface, you get another important and incredibly interesting quantity: Total Curvature.