A gamma-ray burst that occurred "13 billion light-years away" from the Earth
(Gemini Observatory/NSF/AURA, D. Fox & A. Cucchiara (Penn State U.), and E. Berger (Harvard Univ.), apod090429)
The object inside the circle is GRB 090423, the earliest supernova explosion observed as of April 23, 2009. The explosive death of a massive star caused a gamma-ray burst which was aimed almost directly at the Solar System, allowing it to be easily visible despite its light having traveled 13 billion light-years to reach us. Given estimates of the age of the Universe in the range of 13.6 billion years, the supernova occurred only a little over 600 million years after the Big Bang, placing it in (almost certainly) the first generation of massive stars to live and die.
When you read, as in the caption above, that an object is 13 billion light years away and seen as it was only 600 million years after the Big Bang, what does that mean? How is it possible for that object to have gotten so far away in such a short time, when everything presumably started off very close together?
The answer to these questions lies in what we mean when we say that something is 13 billion light years away. In general, such an object was not 13 billion light years away when the light by which we see it left it, nor is it that far away now. What it means is that it took 13 billion light years for the light by which we see it to get here.
For a correct understanding of what is going on we need to refer to relativity theory, which is beyond the scope of this discussion. (Later versions of this page will refer to such relativistic 'corrections' for the benefit of those who would like to see them, but this discussion relies only on a 'common-sense' approach for the benefit of those who just want to understand the basic idea.)
Saying that we see something 13 billion years away, only 600 million years after the formation of the Universe, means that the author of such a statement is using a value of 13.6 billion years as the time since the Universe began. But because of the expansion of the Universe, when we see an object so far out in space that it is seen at nearly the beginning of the Universe, the space between us and the object would have been expanding away from us at (in this case) 13/13.6 or 96% of the speed of light. Any given part of the space between here and there would have been expanding (in all directions) at a relatively slow rate, but with the very large distance between us and the object in question, the total expansion of the space between here and there would have added up to that huge value.
Now according to Einstein's Theory of Special Relativity, light always travels through 'local' space (the space it is currently passing through) at the speed of light. But that does not mean that it is moving toward us at that rate, as we have to take into account the expansion of the space between us and that local space, which in this example is 96% of the speed of light. As a result, even though the light emitted by the burster would have been traveling toward us at the speed of light when it left the dying star (and at every moment since then), its original net progress toward us would only have been at 4% the speed of light (the difference between its speed, and the expansion of space between us and the distant object). As a result, its forward progress toward us would have been 25 times slower than expected, and it could have taken 13 billion years to get here even if the object were only 4% as far away as the stated 13 billion light-year distance (this assumes that the forward progress of the light was always only 4% of its speed through "local" space, and as shown below that is an oversimplification; but as noted at the start, we are proceeding from simpler concepts to more accurate ones, and will get there almost immediately).
The net result is that when the light from very distant objects left them, they were far closer than the light-travel time from there to here. The exact distance they had requires relativistic corrections, because in the example above the light is only slowed by 96% of its progress toward us in the more distant portions of its path. As it gets closer to us the expansion rate of the remaining space is slower and slower, so the forward motion of the light toward us becomes more nearly equal to the speed of light. But although the actual distance the object had when it emitted the light isn't quite the same as implied above, it is still far less than the light-travel time.
It should also be noted that during the time the light was approaching us, the remnant of the object was continuing to move away from us, and in the 13 billion years since it emitted the light by which we see it, it has moved much further away than its original distance. In this example, although far less than 13 billion light-years from us when it emitted the 13-billion-year-old light beam, the star would now be over 30 billion light-years away.
In other words, there are several cosmic distance scales which apply to such a situation. The simplest one to describe, and the one which is therefore generally used, is the time it took the light to get here (in this case, 13 billion years). However, the actual distance the object had when it emitted that light is much less than that value (if it was close to the 'edge' of the observable Universe at that time), and its current distance is much further (and in fact, may be "outside" the observable Universe, so that we may never get to see it as it is now, no matter how long we wait).
Finally, how could this object get hundreds of millions or billions of light years away in the few hundred million years between the beginning of the Universe, and the time the star exploded? Current theory proposes that there was a very short time, just under a millionth of a trillionth of a trillionth of a second, in which the Universe expanded far faster than the speed of light (this is referred to as the Inflationary Theory), so that things could have ended up much further away that even without nearly light-speed motions, once the initial Inflation ended. (But that is another tale, for another time.)
Added Nov 10, 2014: Using the simplest assumptions about the "shape" of the Universe, the distance of GRB 090423 can be calculated from its redshift z = 8.2 (+/- 0.15). This number means that the change in wavelength of spectroscopic features observed in the light of the object is 8.2 times their original wavelength. Calculations based on that number suggest that the object was about 3310 million light years away from us at the time the light by which we now see it was emitted, 13220 million years ago (hence its "distance" of approximately 13 billion light years, as stated in the caption for the image at the top of this page). The difference between its original 3310 million light year distance and the 13220 million years its light had to travel to reach us means that the space between us and it expanded by about 10000 million light years during the light-travel time. Meanwhile, the object was speeding away from us faster and faster, and is now more than 30000 million light years away from us, which means that the empty space between us and it is expanding at a cumulative rate of 2 1/2 times the speed of light, and no light now being emitted by it will ever reach us.
Note that these calculations indicate that the star was over 3000 million light years away from us only 600 million years after the Big Bang. This is one reason for the popularity of the Inflationary Theory. Once the "Big Bang" was over, an object starting near us and traveling away from us at only 96% the speed of light couldn't possibly have gotten 3000 million light years away from us in only 600 million years. But if it was already more than 2500 million light years away from us at the time Inflation ended, it could easily have covered the remaining distance to its observed position in the 600 million years following the end of Inflation.