Euclidean geometry is the geometry of flat space.
It was actually not until the 19th century that it was properly recognized that geometries could exist that did not satisfy all of Euclid's postulates. In Euclidean geometry, parallel lines never intersect. This need not be the case though in non-Euclidean geometries and the behavior of parallel lines serves to distinguish those geometries. There are also other ways to characterize geometries. One is originally due to the Jesuit priest Saccheri in the 17th century.