look-back time
Light is fast but not infinitely fast — it takes time to cross space. That simple fact has a startling consequence: every time you look at something far away, you are seeing it not as it is now but as it was when its light set out. Look at the Sun and you see it as it was 8 minutes ago; look at a galaxy a billion light-years away and you see it as it was a billion years ago. This delay is called look-back time, and it means that to look out into space is to look back into the past.
Concretely, look-back time is how long ago the light we are receiving now actually left its source. For nearby objects it equals the distance divided by the speed of light: a star 40 light-years away is seen as it was 40 years ago. For very distant objects the bookkeeping is subtler, because the universe has expanded while the light travelled, but the core idea holds: the farther away, the further back in time. We never see the distant universe as it is today; we only ever see its history.
Far from being a nuisance, look-back time is one of astrophysics' greatest gifts. It turns telescopes into time machines: by looking at ever more distant galaxies we watch the universe at ever earlier ages, seeing galaxies as infants, the first stars switching on, and ultimately the afterglow of the hot early cosmos itself in the Cosmic Microwave Background. The honest flip side is that we can never know what a distant galaxy looks like 'right now' — that information has not had time to reach us, and by the rules of physics, it never can while it is far enough away.
The Andromeda galaxy is about 2.5 million light-years away, so the smudge of light you can see with the naked eye on a dark night left it 2.5 million years ago — before modern humans existed. You are quite literally looking at deep prehistory.
To see far is to see long ago — distance in space is distance in time.
A light-year is a distance, not a time — but look-back time converts that distance into the time since the light departed. Saying a galaxy is 'a billion light-years away' is the same as saying we see it as it was a billion years ago.