Space form

pseudosphere

The pseudo sphere is a regular surface with the the peculiarity of having constant Gaussian curvature equal to -1.

It is not simple connected surface.

Zero curvature surface different to the plane

Sphere

The unitary sphere is a regular surface having constant Gaussian curvature equal to 1.

There is a beautiful theorem that tells us about a classification of connected, simply connected, complete Riemannian manifold of constant sectional curvature. Under these hypotheses, the spaces of constant curvature are classified up to isometry by the value of their curvature.
Therefore, we said that a Riemannian manifold with constant sectional curvature is elliptic, hyperbolic or flat depending on whether its sectional curvature is positive, negative or zero.

Reference: Kobayashi, S., Nomizu, K. (1963). Foundations of Differential Geometry. Reino Unido: Interscience Publishers.(Google Book)
(cf.Theorem 7.1 of Chapter VI).