A moment's thought (or some careful examination of maps) reveals that any circle drawn on a globe is in fact a circle in real life. The same claim holds for circles drawn on flat surfaces (obviously).
Are these the only examples?
Formally, let $M$ be any Riemannian surface, (isometrically) embedded in $\mathbb{R}^n$. Moreover, suppose that any metric circle $C$ (i.e., a locus of points with constant geodesic distance from some fixed point) is in fact a circle in $\mathbb{R}^n$. Then must $M$ have constant nonnegative Gaussian curvature?