When asking the question "How many stars are there in the universe?" it's important to distinguish between the universe as a whole and the observable universe. Because the universe was born 13.8 billion years ago, we can only observe objects up to a certain distance from Earth — light from more distant objects hasn’t had time to reach us yet. And to answer “how many stars are there,” we must limit the discussion to what we can observe.
Astronomers had estimated that the observable universe has more than 100 billion galaxies. Our own Milky Way is home to around 300 billion stars, but it’s not representative of galaxies in general. The Milky Way is a titan compared to abundant but faint dwarf galaxies, and it in turn is dwarfed by rare giant elliptical galaxies, which can be 20 times more massive. By measuring the number and luminosity of observable galaxies, astronomers put current estimates of the total stellar population at roughly 70 billion trillion (7 x 1022).
However, those estimates are dependent on the sensitivity of current telescopes. More recent estimates have upped the number of galaxies in the observable universe to 2 trillion, though many of these are tiny, fluffy galaxies with fewer stars. If the typical galaxy had 100 billion stars, then there would be 2 x 1023 stars in the observable universe, three times as much as earlier estimates. But even this is still probably an underestimate, as more sensitive telescopes will continue to reveal fainter galaxies and stars.
How many stars are there in my night sky?
The number of stars you see in the night sky depends on several variables, including your location’s light pollution and your own vision. In large, light-polluted cities, only a few dozen of the brightest stars may be visible - though that doesn't mean there's nothing to observe from a city. But in a clear, dark sky, a couple thousand stars become visible to the unaided eye.
If you tabulate all stars visible down to magnitude 6.5, thought to be the faintest stars still visible to the unaided eye, the entire sky contains some 9,000 stars. Since you can only see half the sky at any time, that means there are as many as 4,500 stars visible in your sky tonight.
Read more on where this number comes from in Bob King's article, "9,096 Stars in the Sky — Is That All?"