Hyperbolic geometry is an example of a geometry where the parallel postulate fails in the sense that given a line L and a point P not on that line there are infinitely many lines through P that are parallel to L.
The parallel postulate stood out from Euclid's other postulates for geometry in that it was less elementary than the others.
For almost 2000 years people attempted to resolve this by showing that it followed from the other axioms of geometry. This finally came to an end in the beginning of the 19th century. A model of geometry where the parallel postulate does not hold was claimed by Lobachevsky, though later it was learned that Karl Friederich Gauss had come up with one earlier.