There are three basic types of geometry: Euclidean, hyperbolic and elliptical. Although there are additional varieties of geometry, they are all based on combinations of these three basic types. Euclidean geometry is the original form, dating back to 300 BC, and it is the result of the work of the Greek Alexandrian mathematician Euclid, who developed the five postulates, or axioms, upon which his geometric theorems are built.
Euclid's fifth postulate, known as the parallel postulate, went without an accompanying proof for thousands of years. The parallel postulate assumes that a straight line crossing through two other straight lines and forming two same-side interior angles of less than 90 degrees determines that those two lines, if extended far enough, will eventually meet on the side of the interior angles. This assumption, however, did not take into account the idea of curved space, which was first conceptualized by Albert Einstein in his 1915 General Theory of Relativity.
The idea of space existing with either a positive or a negative curvature introduced the idea of non-Euclidean geometry, in which the parallel postulate would not always hold true. In curved space, it cannot be assured that the two lines in question will ever meet, regardless of how far they might be extended. The geometry based on space with a negative curvature became known as hyperbolic geometry. Elliptical geometry refers to the type of geometry based on space with a positive curvature.