One way of thinking of this is to consider what would happen if
the sun were to suddenly wink out of existence.
As far as I understand it, one of the big differences between Newton's and Einstein's theories of gravity is in their treatment of this question.
Under Newton's theory, the earth changes path immediately and heads off in a straight line, tangent to its previous orbit (modulo the slight effects of other planetary bodies.)
Einstein, however, thought that the earth's path would not diverge for about 8 minutes (or whatever time it takes light to reach the earth from the sun.) This was (if my memory is correct) actually one of the considerations that led him to formulate his theory of relativity.
At any rate, the standard formulation of general relativity assumes this is true, and that gravitational effects propagate at the speed of light.
(update: 8/9/2002)
Although GR is one of the most well tested scientific theories, and it has passed every test so far, this aspect of the theory has not yet been tested - but it is due to be tested today (Sept 8, 2002). Physicists hope, by measuring the effect of the gravity of Jupiter on light from quasar QSO J0842+1835 (a very distant object), which will pass within about 3.7 arc seconds' angular distance from it, to resolve the question once and for all.
The perturbation of the light will be measured using "Very Long Baseline Interferometry" (VLBI) as proposed by Sergei Kopeikin in his letter Testing the propagation of gravity by very long baseline interferometry in Astrophysical Journal, 556 (1): L1-L5 Part 2, Jul 20 2001, where he says:
we emphasize that there are two relativistic parameters to be measured in the VLBI experiment in order to test the validity of general relativity -- the PPN parameter gamma and the gravity propagation parameter delta [...] The primary goal of the new experimental test of general relativity proposed in the present Letter is to set direct observational limits on the parameter delta that will measure the effect of retardation in the propagation of gravity by the moving Jupiter. According to the Einstein theory of relativity, one must expect that the numerical value of the parameter delta must be equal to zero.
Got that?
Me either. I guess what it means is that the exact timing of the wobble in the quasar's light caused by Jupiter's gravity, as Jupiter moves towards and past the quasar's position in the heavens, can be used to determine whether Jupiter's gravity is propagating as predicted by the theory.
Kopeikin himself has indicated he expects delta will be measured at zero, and most physicists will surely be very surprised if general relativity fails this test where it has passed so many others - the expectation is likely to be along the lines of "Ok, Einstein was right again. Back to work, everyone."
However, if the result diverges from the predictions of GR, and can be confirmed, physics will take a big step into uncharted territory. A theory will be required that makes all the predictions made by general relativity, except the predictions about frame dragging, which is where the propagation speed of gravity enters into the theory. Such theories are currently either very obscure or nonexistent. Additionally, the fortunes of gravitons - the so-far undetected particles which are hypothesised by particle physicists to transmit the gravitic force, as photons transmit the electromagnetic one, as indicated below, will surely take a severe turn for the worse.
Info about Kopeikin's proposal from the usenet newsgroup sci.physics.research, in the thread "Speed of gravity" -- A final resolution and, in particular articles <ajl94l$c2n$1@glue.ucr.edu> by John Baez and <Pine.GSO.4.42.0206160810080.27740-100000@clyde> by Stephen Speicher.