Elementary, my dear self.

     Non-Euclidean geometry is defined as any geometry in which the parallel postulate is not true, which tends to occur in elliptical and hyperbolic planes (as opposed to flat ones, of course), hence the two main geometries within the non-Euclidean ones (geometry-ception, eh?), aptly named elliptical and hyperbolic geometry. The parallel postulate can refer to Euclid's fifth postulate (If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles) or Playfair's axiom (Through any point in a plane, there is at most one straight line parallel to a given straight line).