Astronomers describe the darkness of the sky under which they are observing by referring to the limiting magnitude, that is the magnitude of the faintest star we can see.
We might use a very broad brush approach, and describe a sky as “fourth mag” (roughly what you’d see from the suburbs of a big city) or “sixth mag” (very dark indeed, with almost all but the very faintest stars visible).
The absolutely faintest stars anyone can see with the naked eye in perfect conditions (i.e. looking towards the zenith on a moonless night) are around 8th mag (this is for someone with exceptional vision), and so we refer to a perfect sky, utterly free of light pollution, as being magnitude 8.
However, most star maps and charts will only list stars down to a more modest limiting magnitude of 6.5, which number is often used as the standard limiting magnitude of a very dark sky.
The limiting magnitude of a sky has implications for how many stars you can see with the naked eye, and as a very rough rule of thumb for every one point of magnitude darker your sky is, you will see three or four times as many stars.
There are around 50 stars of magnitude 2 or brighter, which will therefore be visible in even the most light polluted skies with limiting mags of 2. Of course these 50 stars are spread over the whole sky so you’ll only ever see around half of them at once, and so in a very light polluted place like Manhattan, Las Vegas or Tokyo you can see around 25 stars only.
In the centre of other big cities, where the limiting magnitude is 3 you will see roughly 100 stars, and from the suburbs under magnitude 4 skies you will see around 300 stars (these star counts, and the ones that follow, are for half the sky only, i.e. in your visible hemisphere)
Once you get to a limiting magnitude of 5 in rural areas with a bit of light pollution you can expect to see 1000 stars or more. Further from big towns and cities, where your sky has a limiting magnitude of 6 you’ll see 3000 stars. At the limit of most star maps and charts, limiting mag 6.5, there are around 4500 stars to be seen in the sky at any one time, and so there are around 9000 stars altogether – over the whole sky – that are magnitude 6.5 or brighter. (As a little aside, the planetarium that I used to run in Glasgow Science Centre, had a Carl Zeiss starball projector that claimed to show 9000 stars, and so was obviously built to show every star that would feature on any standard star map or chart).
And beyond that you’ll need a telescope or binoculars to see more. Remember, those 9000 stars are only a tiny fraction of the 100,000,000,000 stars in our galaxy!