The parallel postulate (aka the fifth axiom) doesn't work in spherical geometry, in the most dramatic way: the "lines" (geodesics) are exactly the great circles, or intersections of planes going through the centre of the sphere with the sphere. So any two great circles belong to two planes sharing a point (the centre of the sphere!). But two planes in R3 sharing a point must share a straight line, which intersects the sphere at two antipodal points. Thus every two great circles intersect at two antipodal points; parallels are simply impossible here!

Actually spherical geometry just manages to violate another of Euclid's axioms. Between every two points there's a "line" (a shortest path), but it's not always unique! Consider two antipodal points (again). Then there exist infinitely many geodesics between them, and "both ways" on each geodesic have the same length. There's definitely more than just one shortest path between antipodes!