What does it mean to say that space is curved, and how do we measure it?
It's simpler to look at things in two dimensions. Imagine a flat piece of paper. How would we measure its flatness if we were living on the paper, like ants, not able to perceive a third dimension? There's a neat way of doing this. Get a ruler, and mark out a line. Now move the ruler around, always keeping the ruler pointing in the same direction relative to yourself. Move back to the marked out line. If the ruler is still parallel to that line then the surface you are on is not curved. If the ruler is no longer parallel to the line then you have moved over a curved surface. The ruler has been twisted - not by you, but by the curvature of space.
We can do exactly the same in three dimensions. We use a ruler, but this time, a line is marked in three dimensions. Move the ruler around, keeping it pointing in the same direction relative to yourself. Now move back to the original line. If the ruler and the line are not parallel then you have moved through curved space!
Curved space is a prediction of Einstein's General Relativity, and we can actually measure its curvature using very, very precise gyroscopes, which remain pointing in the same direction. If such gyroscopes are put orbit around the Earth and space is curved and twisted by the Earth, the gyroscopes, like the imaginary ruler, will also twist. This has been done! http://en.wikipedia.org/wiki/Gravity_Probe_B