Any point on a smooth surface can be uniquely classified by its Gaussian and toric curvatures. All other points with non-zero toric curvature are called toric points . Elliptic points look locally like an outside equitorial patch on a torus, curving more in one direction that the other.
Parabolic points look locally like a cylinder, flat in one direction and curved in the other.
Hyperbolic points look locally like an inside equitorial patch on a torus or a saddle, rising in one direction and falling in the other.