Euclid's geometry was based on five postulates (assumptions). The first four postulates are simple, clear and logical. The fifth postulate has always been a problem. It is a lot more complicated than the first four. It can be stated in many ways, but it basically says that parallel lines never meet. For centuries, mathematicians tried to prove the fifth postulate from the first four. In the 19th century some mathematicians started a new approach. They looked at what would happen if the first four postulates were true but the fifth was untrue. Their investigations led to non-Euclidean geometries. The two basic forms of non-Euclidean geometry are elliptic geometry--where parallel lines always meet--and hyperbolic geometry where parallel lines diverge.
Elliptical geometry is the geometry of surfaces with positive curvature. On surfaces with positive curvature parallel lines always meet. Another characteristic of spaces with positive curvature is that the sum of the angles of a triangle is always more than 180 degrees. An important example of a non-geometry is the surface of the earth. On a sphere, straight lines are called "great circles." They are circles of maximum size that divide the sphere into hemispheres. On the earth's surface, the equator is the only line of latitude that is a great circle. The other lines of latitude are not great circles and so not "lines" in non-Euclidean geometry. All of the lines of longitude are great circles, so they are all non-Euclidean "lines" and they all meet at both of the poles.
Hyperbolic geometry is the geometry of surfaces with negative curvature. Euclidean geometry is the geometry of space with no curvature at all. On surfaces with negative curvature parallel lines always diverge. Another characteristic of spaces with negative curvature is that the sum of the angles of a triangle is always less than 180 degrees. It is not so easy to find a real world example of a hyperboloid space although it has shown to be self consistent for certain abstract mathematical models such as the tractricoid.