Or is it everywhere around it?
It IS everywhere, and as Dave posted, and does extend asymptotically to infinity...BUT as the illustrations above imply, the curveature is much less prounced at greater distances... what that means, of course, is that if a spaceship passes close to Earth at some firxed velocity it's path is affected a lot more than if it were to pass by but,say, in another galaxy.
As another example, the massive black hole believed to be at the center of our own galaxy gobbles up lots of stuff nearby...but Earth at it's great distance is barely affected... eventually we might end up there too, except that probably our sun will engulf the Earth (erupting as a red dwarf) first and ruin all the fun.
Purists in these forums often HATE those diagrams...but I believe they are a good place to start recognizing their limitations...
Maybe a good next step is to look at world lines in special relativity...no curvature, no gravity:
http://en.wikipedia.org/wiki/World_line
A major problem is that no one can visualize curved space..let aloe curved time...nor the way that space and time become intermingled in general relativity...they partially change into each other...One way to begin to understand some of that is to note that in a gravitational field, say of a planet, where space time curvature is prominent, time slows relative to a distant observer...gravitational potential affects perceived passage of time as does velocity...!
Both gravitational potential (GR) and velocity (SR) affect perceived spacetime...the latter via length contraction and time dilation...
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