This is a question which plagued Scientists for a very, very long time. Initially posed by a man called Olbers (I can't remember his first name, I've got a feeling it was William?) The idea is that because there are so many stars, in all directions, all giving out light...why isn't the sky glowing with star light, why is it mostly dark?
The quick answer is that there are a finite number of stars and the Universe hasn't been around forever. So we can see all the stars which have formed, but only a certain number have formed and, given how small we are compared to the Universe, we're not going to see most of the stars because they're so spread out and far away. But this doesn't get to the real heart of the question: In the past, the Universe must have been filled with light from the big bang's initial expansion, the formation of atoms and the formation of early stars...where has all this light gone?
The answer to this question, rather surprisingly, is that the night sky IS filled with starlight, it's just our eyes which can't pick it up. Our Universe seems to have started existing in its present state around 13.7 billion years ago. Stars don't form instantly, in fact they take millions of years to grow and start glowing. So in the initial Universe there were no stars, but then they started forming and giving light out.
That light still exists in the Universe and it's flying about everywhere. The only problem is that the Universe is stretching. For reasons which aren't understood yet, our Universe is expanding at such a rate, that beams of light get stretched out with it. What we know is that light is, in a sense, made of ripples in the electromagnetic field all around us. As this field gets stretched by the Universe (that's the bit we don't get) the beams of light are stretched too.
As the beams of light get stretched, their colour changes. Suppose a Sun is glowing yellow, that beam of light is yellow when it emerges from the Sun, but is gradually stretched to orange, then red, then infra-red, then microwaves and so on. This means that the really old stars gave their light out so long ago, and it's been stretched so much, that by the time it reaches us, it's been stretched beyond the visible spectrum.
The more recent stars' light hasn't been stretched as much, so we see it visibly as (usually) white light. If we could see infra-red light, then the sky really would be glowing because we'd be able to see billions and billions of stars in all directions. So ultimately, the reason the sky appears dark is down to three things: 1) The Universe is finite in time. 2) The Universe is stretching. 3) Stars take time to form.