To the unaided human eye, the night sky appears dark because, from the perspective of an individual on the surface of the planet, "night" occurs when that part of the planet is facing away from the Sun. Our eyes evolved - or were designed, according to some - to function normally given the level of light we receive from our own Sun during the day. Were our eyes far more sensitive, than daylight could be blinding, even painful - much like the sensation one gets from turning on bright lights just after waking up in the morning. The down side is that our night vision is quite poor - and the sky above us appears dark.
In truth, of course, the night sky is not truly "dark," in the sense that "dark" means no light whatsoever. The night sky is actually a rich source of visible light and electromagnetic radiation. Starlight and, much more importantly, the Moon actually provide enough light that on the surface we can see outside and navigate using the faint glow. People living in cities will notice that the night sky is never truly dark, particularly when it is cloudy out: building lights and streetlights always provide some light, which reflects off the clouds above.
Still, on a cloudless night in the countryside, the sky can indeed appear completely dark to the naked eye. The Moon appears bright because it is reflecting sunlight toward us. Moreover, many of the stars in the sky are actually far larger and more luminous than our own Sun. Still, all of these are extremely far away. The nearest star to our own, Alpha Centauri, lies hundreds of thousands of times as far from Earth as our own Sun does. In fact, the bright stars are so far away that the brightest points in our sky are not stars at all, but planets, like Mars and Venus, which are reflecting sunlight in the same way as the Moon. None of the other stars are close enough, and none of the planets large or mirror-like enough, to make up for the loss of sunlight. For this reason, to us, the night sky always appears dark.
Still, this is largely a reflection of the limitations of our own eyes. Most of the stars are so distant that we can see them only with telescopes; indeed, some are so distant, billions of light-years away, that we tend to observe whole galaxies rather than single stars. The deep field images produced by the Hubble Space Telescope reveal a sky much richer in stars, galaxies, nebulae, and other massive formations than we could ever see with the naked eye. Many areas of the sky that appear completely dark to us on the surface are actually lit up with stars and galaxies, but so faintly that we cannot see them unaided.
Even with a powerful telescope, however, much of the sky between the stars and galaxies is still dark in visible light. Despite the enormous size of the universe, this means that in that precise direction there are no objects large enough and bright enough for us to detect them, within the limits of the observable universe. Whether there are still more objects beyond the observable universe, i.e. beyond the limits of light which has been able to reach us since the origins of the universe (roughly 14 billion years), is uncertain, although we assume that the universe extends beyond what we can see.
Finally, there is one more reason that the night sky would not appear dark if we could see beyond the limits of visible light. When the sky is scanned at microwave frequencies, rather than at visible light frequencies, the entire universe, even the patches of sky that are dark to the naked eye, have a faint glow to them, which scientists refer to as cosmic microwave background radiation. It is believed that the cosmic microwave background is the result of reactions which occurred in a very early phase of the universe's history.
However, without powerful telescopes or microwave detectors, for the majority of us the night sky will always appear mostly dark. This is because, to the human eye, none of these sources - distant stars, cosmic microwave radiation, reflecting planets, or the surface of the Moon - can direct enough light to the dark side of the Earth to make up for the loss of sunlight.