Determining the number of stars in the sky is actually a surprisingly difficult problem, because it leads directly into a second problem: determining the size of the universe. The European Space Agency says there are between 10 sextillion and one septillion stars in the universe (respectively a number 1 followed by 22 zeroes, and a number 1 followed by 24 zeroes). The fact that one of the world's premiere space organizations is effectively admitting their estimate may be off by as much as a factor of one hundred indicates how unsure people are. However, an Australian survey done in 2003, which tried to be more specific, came up on the low end of this spectrum: 70 sextillion (7 followed by 22 zeroes).
On any given clear, dark night, of course, the human eye cannot detect the vast majority of these stars. On a good night in a rural area, you might be able to make out as many as several thousand stars. In a city, where ubiquitous lights on the ground ruin night vision and obscure the sky, you would be lucky to make out a fraction of those, probably a few hundred of the brightest stars. However, this "number of stars" is somewhat misleading. Some of the apparent stars are actually distant galaxies. Others, including some of the bright ones, are actually two or more stars orbiting closely together. The brightest star visible from Earth, Sirius, is just such a binary system, consisting of one large white main-sequence star (brighter than our own Sun) and one white dwarf, the glowing ember of a dying star which has exhausted its hydrogen resources.
Simply counting the number of stars in our own Milky Way Galaxy is a more daunting task. Not even all of these stars can be found, even with powerful telescopes: when one peers directly along the galactic plane, gas and dust in the interstellar medium means it becomes impossible to make out actual stars beyond a distance of about 6000 light-years. Moreover, dim red dwarf stars would be hard to make out at significant distances, too, at least with currently operating technology. However, astronomers currently believe that the Milky Way Galaxy is home to around 100 billion stars, though the real number may be as high as 400 billion.
Around the Milky Way Galaxy orbit a handful of so-called dwarf galaxies, small galaxies with only a fraction of the number of stars. However, among the major galaxies, the Milky Way Galaxy is at best average; our nearest neighbour, the Andromeda Galaxy, is several times as large. About thirty of the nearest galaxies are grouped together as the so-called Local Group, which is, in turn, one outpost of a much larger formation of galaxies called the Local Supercluster. It is believed that the universe contains many millions of such superclusters.
Obviously, when dealing with such enormous spans of distance and matter, determining the actual number of stars starts to become a pretty daunting task - as well as seemingly increasingly meaningless. The Australian survey's estimate - 70 sextillion - was based on their best effort to take a tiny slice of the observable sky, count all of the stars within that small area of space, and then extrapolate that to the entire night sky, assuming that everywhere else the stars will be distributed in comparable numbers. Actually counting the stars in the entire sky, as seen through today's most powerful telescopes, would be a mammoth task, requiring many lifetimes of human labour and/or an intimidating amount of computer processing resources.
However, even that search soon bumps up against another difficult problem, which is more complex than it first sounds: how big is the universe? Light from the edge of the universe would take billions of years to reach us. Because the universe is only about 14 billion years old, however, if there was anything more than 14 billion years away from us, we have not yet been able to see it. The situation is further complicated by the fact that the objects which were 14 billion light-years away 14 billion years ago, when they emitted the light that is finally reaching telescopes on Earth, have been travelling away from us for that entire time, meaning they are now actually much farther away.
The end result is that we can be relatively sure that there are, in the observable universe, millions of superclusters consisting of billions of galaxies, consisting of sextillions of stars. However, we will never be able to count all those stars, and we will never be sure if there are even more beyond the limits of our vision.