If you cut an orange in half, scoop out the inside, and try to flatten one of the resulting hemispheres of peel, then you will tear it. If you try to flatten a horse’s saddle, or a soggy potato chip, then you will have the opposite problem: this time, there is “too much” of the surface to flatten and you will have to fold it over itself. If, however, you have a roll of wallpaper and wish to flatten it, then there is no difficulty: you just unroll it. Surfaces such as spheres are said to be positively curved, ones with a saddle-like shape are negatively curved, and ones like a piece of wallpaper are flat.
Notice that a surface can be flat in this sense even if it does not lie in a plane. This is because curvature is defined in terms of the intrinsic geometry of a surface, where distance is measured in terms of paths that lie inside the surface.
There are various ways of making the above notion of curvature precise, and also quantitative, so that with each point of a surface one can associate a number that tells you “how curved” it is at that point. In order to do this, the surface must have a RIEMANNIAN METRIC [I.3 §6.10] on it, which is used to determine the lengths of paths. The notion of curvature can also be generalized to higher dimensions, so that one can talk about the curvature of a point in a d-dimensional Riemannian manifold. However, when the dimension is higher than 2, the way that the manifold can curve at a point is more complicated, and is expressed not by a single number but by the so-called Ricci tensor. See RICCI FLOW [III.78] for more details.
Curvature is one of the fundamental concepts of modern geometry: not only the notion just described but also various alternative definitions that measure in other ways how far a geometric object deviates from being flat. It is also an integral part of the theory of general relativity (which is discussed in GENERAL RELATIVITY AND THE EINSTEIN EQUATIONS [IV.13]).