In the context of undergraduate multivariate calculus courses, a saddle point is a point on a two-dimensional surface in three dimensions that is a local minimum in one direction and a local maximum in another. More generally, it's a point on a differential manifold where the Gauss curvature is negative. Without going into the mathematical details, one can think of it as a point where the surface is both concave and convex, like a saddle: you can sit *in* it and *on* it at the same time.