The parallel line postulate is a postulate for good reason; Euclid himself suspected it, but was unable to construct an argument against it. It took much later, as mathematicians were toying with complex, nonlinear spaces, for the postulate to finally be questioned, and for alternatives to apply.

Spherical geometry is such a case: the angles of a closed polygon always sum to less than the angles of the analogous figure on the flat plane, and parallel lines always intersect. The opposite case is hyperbolic geometry, where the angles sum greater than in the flat plane, and parallel lines always diverge.

The most extreme example, on a sphere, is to travel in a straight line for half the circumference, turn 360 degrees, and travel the same again. In the plane, this describes only a line segment, or at best, a triangle with one side zero, either way enclosing zero area.

Hyperbolic geometry isn't all that unfamiliar. Many naturally frilled and crumpled shapes exhibit this. Lettuce leaves and decorative frills are examples. The construction is simple: suppose you create a flat circular rug by sewing material onto the perimeter of an initial nucleus; but each course of material you add, is scrunched up so that it is longer than the actual perimeter it's being added to. It crumples up as you go, and the perimeter grows disproportionately to the distance from center (that is, you make the perimeter increase faster than pi times the diameter). Well, that sounds rather shoddily constructed, doesn't it -- it won't sit flat! But the benefit is packing a lot of surface area into a small volume, while making it easily accessible: that is, within a short distance from the center (depending on how much larger you made "pi" in this object).

On a frilly surface, because the circumference of a circle increases rapidly with distance from center -- that's literally how it's sewn together, eh? -- you can imagine moving from the origin out to some radial distance, then following a tangent and returning to the origin, and having subtended only a small angle at the origin. That is, if you draw triangles around the origin, you can pack many more than, say for example, six equilateral triangles around the origin!

Which leads to another construction method: suppose you join equilateral triangles by their edges. YOu can choose how many to join per vertex. Six gets you the flat plane (with a triangular / isometric grid), five or fewer gets you a somewhat spherical section (the icosahedron; four gets you an octahedron and so on!), seven or more gets you a hyperbolic frill. If you count the grid distance (number of triangles) traveled between points and along paths, you get the same results as before.

Now, on Earth, or on the surface of any round-ish object, really: the curvature is a consequence of that object's surface, and the confinement of activity to its surface (because we can't really swim through the Earth's crust, and it's rather hard to fly arbitrary heights above its surface). This is not at all due to relativity.

Relativity does have an effect on geometry, but it's only significant for extremely precise calculations (e.g. GPS), extremely massive bodies (solar masses and up), or extremely high speeds (near the speed of light). In this case -- now, let me see if I have this right? -- hyperbolic space is present locally around a gravity well, or a property of the universe itself if the matter density is a little too low (which doesn't quite seem to be the case), while spherical geometry is present inside a gravity well of sufficient size: a black hole's event horizon, or a universe with slightly too much matter (which also doesn't quite seem to be the case). You also can't move or see beyond your causal horizon, which is also the extent of the visible universe if the universe happens to be flat (which seems to be very nearly the case).

If one constructs an experiment to measure the distances and angles in a triangle, say -- then the geometry of that experiment is affected by fluctuations in space itself: gravity waves. A number of active and proposed experiments are working on this very problem, and they've shown exciting (if extraordinarily weak) signals already!

Tim