Non-Euclidean geometry is a form of geometry that rejects Euclid's fifth postulate that parallel lines extend to infinity and never intersect. There are two main forms of non-Euclidean geometry, hyperbolic, and elliptic. The former allows all lines to intersect, and the latter has non-intersecting lines that diverge in both directions. In Euclidean geometry the sum of angles in triangles is always exactly 180 degrees, in hyperbolic geometry triangles always have less than 180 degrees, and in elliptic geometry triangles always have greater than 180 degrees. The roots of Non-Euclidean geometry were in works by Gauss, and his pupil Riemann. Later nineteenth century mathematicians such as Janos Bolyai and Nikolai I. Lobachevsky developed this considerably.
A geometry is defined as the pure mathematics of points and lines and curves and surfaces. It works by defining a set of axioms which clarify how points and lines are constituted, for example: two distinct points define a unique straight line, and deriving from these results. Some geometries such as the Fano Plane many have seven points or fewer. The simplest geometry is the empty set.
An easy way to visualize elliptic geometry is to imagine lines drawn not on a sheet of paper, but instead on the surface of a sphere. Consider a "triangle" drawn on the globe, starting at the North Pole, then south along the Prime Meridian to the equator (a point somewhere off the west coast of Africa), then west along the equator to, say, the point where the equator meets Africa, then north along whatever line of longitude that is back to the North Pole. We not have a "triangle" composed of three "straight" lines, but the angles of the triangle add up to more than 180 degrees. To see that, consider the "base" of the triangle along the equator. Since lines of longitude intersect the equator at right angles, the angle of each side of the base of our triangle is 90 degrees and together 180 degrees. Since the angle at the North Pole is not zero, the triangle must contain more than 180 degrees.