Measuring the speed of light and defining the metre - absolute or relative?

There are three relevant quantities involved here: the length of a meter, the duration of one second, and the speed of light. You only need to absolutely measure one of them, after which the other two can be defined in terms of the one that is measured.

For technological reasons, we have chosen to make the measured reference quantity the length of one second, which is defined in terms of the number of oscillations of radiation associated with the transition between the hyperfine ground states in cesium (specifically, it's 9,192,631,770 oscillations of that light). This is basically because there are experimental techniques that allow incredibly precise measurements of the frequency of radiation, at a level that really can't be matched by length or speed measurements. (The best frequency measurements in the world use trapped aluminum ions as the "clock," and are good to something like one part in $10^{18}$.)

Having defined the second in terms of some physically measurable quantity, we are then free to define the speed of light as having some particular value in meters/second, and then define the meter in terms of the distance traveled by light in one second. The size of a meter is merely a matter of convention, not anything fixed in the physical world, so as long as we have anchored the second to something fundamental, we can make the meter be whatever we want.

The particular values of the meter and the speed of light that we choose are based on older measurements using a meter defined in terms of the circumference of the Earth. We've chosen to keep that value, because it would be a hassle to make a wholesale change.

Basically, what happened is that we managed to measure time very accurately with the advent of atomic clocks. Einstein's theory of relativity predicts, or rather posits, that the speed of light is an invariant. So measuring time accurately implies we can measure distances accurately too. We only have to check that light is indeed an invariant and there are also easily reproducible and precise lab techniques to check that, for instance by interferometry. This allows then to define the meter. The number 299,792,458 is then chosen so to make the meter agree with the old definition which used a standard rod.

It seems that in practice, the meter is often determined by measuring wavelengths of certain atomic transitions, read further on:

http://en.wikipedia.org/wiki/Metre

I just complete @Chad Orzel's response by point : you seem to be worried on the effect of chosen frame of reference and space-time curvature. One of the reasons to redefine the meter in term of speed of light is because the speed of light is independent of the frame of reference and the space time curvature, while distance and time are not. That the whole point of Einstein relativity theories.