From a little tidbit by an interested informer, I got to learn about someone wondering if the measured distances of stars and galaxies are really applicable when it takes so long for their light to reach us, that their actual distances now could be completely different. It is actually a bit mind-bending to think about all the issues, which happen to be the sort of thing I love to think and explain, except my explanation was getting rather long for a comment.
For relatively close objects (the Andromeda galaxy would count as relatively close in this context!), the notion of a distance is pretty clearcut. The observed distances should pretty much be the distances now, unless we're missing something drastic that has happened in the meantime. So for Alpha Centauri, whose light took about four years to reach us, it should still be about four light-years away, especially since it's been that way four years ago, and the same way four years before that, ad semi-infinitum. It's certainly possible, although very, very unlikely, that it might not be four light-years away now, perhaps because of a close encounter with an unseen star or even dense clumps of dark matter, or it might not even be there anymore as a victim of a Neutronium Alchemist. But fanciful speculations aside, there's no reason to think that Alpha Centauri would not be four light-years away now.
Even the Andromeda Galaxy, which is about 2.5 million light-years away, would pretty much be 2.5 million light-years away now despite the light having taken 2.5 million years to reach us. We have a pretty good idea of how gravity works, and with the closest object that could affect the galaxy's movement significantly being our own Milky Way Galaxy, it's pretty certain that it is even now approaching our galaxy at a steady 300 kilometers per second. Unless photino birds have flung the Andromeda Galaxy, or even our own galaxy, as a giant projectile weapon against a million light-year wide artifact ...
Things get messy for objects that are really far out. I'm pretty sure that in most cases astronomers refer to distances by how far the light had to travel. However, this is only the distance that the light itself had traveled. We can also think about the actual distances to an object when the light was emitted and when we received the light. Because of the expansion of the universe, the distance between two objects grows as light travels between them. This is not much of an issue for objects that are relatively close since the space between them expands only a little compared to the distance between them, but the changes are rather large for objects that are very distant on a cosmological scale.
For example, between the Milky Way Galaxy and the Andromeda Galaxy which are a mere 2.5 million light-years apart, the expansion of space will increase the distance between the galaxies by less than 10,000 light-years in 2.5 million years, so it would still be about 2.5 million light-years apart even if the expansion of space were to be taken into account. (In reality, this is more than canceled out by the movement between the two galaxies due to the gravitational attraction.) In contrast, something that is 10 billion light-years away from us in terms of how far the light had to travel was actually about 6 billion light-years away when the light was emitted while it would be about 16 billion light-years away right now, assuming nothing drastic happened to the thing in the meantime.
But even this glosses over some thorny issues. What do distances then and now really mean? And are they really meaningful? There isn't going to be any measuring tape in the universe which can measure distances of billions of light-years, and the expansion of space means that a most suitable measure for short distances, the travel time of light, is no longer the unambiguously clear distance measure it could be. There can be alternate ways to measure a distance at a specific time such as having observers about every million light-years apart, waiting until the universe has aged some exact amount of time, say by agreeing to wait until the redshift of the cosmic microwave background is some specific value, then measuring the distances between the closest observers and signaling the measurements among each other, which could all be added up to be the distance at some specific time during the life of the universe. It would take billions of years, but it could work.
Obviously, this is way beyond practical for us puny humans, so we have to infer the distance just from the light that arrives at Earth. But while we could still do a reasonable job of measuring how far the light must have traveled using standard candles, inferring the distances then and now depends on how well we can model the universe. But even if we can compute these distances accurately, which we can do a much better job of thanks to data from WMAP (although I still have little idea of how the fluctuations in the cosmic microwave background are supposed to help), the distances then and now may not be the best ways to express how far distant cosmological objects are.
The problem with the "current" distance is that what we see of a distant cosmological object is what it was a long time ago, not what it is now. For a quasar that is about 10 billion light-years away in terms of light travel distance, the object would now be at about 17 billion light-years away. But the problem is that the object is by now a very different object than when it was a quasar. In fact, it would almost certainly be an ordinary neighborhood galaxy by now. It would be kind of awkward to say that the quasar is 17 billion light-years away when it was not 17 billion light-years away during the time it was a quasar, while it is no longer a quasar now when it really is 17 billion light-years away.
The "old" distance, the distance to the object when the light was emitted, is even more problematic. Space had not expanded as much when the universe was young so that everything was that much closer. In fact, depending on when the light was emitted, the distance to the object could have actually been smaller at the time despite the light traveling even greater distances. For an object about 13 billion light-years away in terms of light travel distance, it could have really been about 3 billion light-years away at the time the light was emitted. Meanwhile, an object about 8 billion light-years away could have really been about 5 billion light-years away at the time the light was emitted. An object that was closer when it emitted the light can actually be farther away!
And I'm not even going into the fundamentally different ways that distance could be defined over cosmological scales, not to mention what "then" and "now" might really mean in our relativistic universe. Given all the ambiguity and uncertainties, astronomers much prefer to express distances to distant cosmological objects with something they can directly measure, the redshift of the spectrum. Unfortunately for most ordinary people, it might not mean much that the redshift is 0.1, 1.0, or 10, so we usually get distances in terms of how far the light travelled. One billion, 8 billion, or 13 billion light-years are somewhat more comprehensible to a layperson like me, which I usually accept without much thought. But I get a headache when I really think about all the issues that surround cosmological distances.