|MadSci Network: Astronomy|
This is a difficult question, especially as we don't know the real nature of the universe, we can only speculate.
First off, let's bypass the arguments about whether the universe really is expanding or not. There are plenty of astronomy sites on the net talking about that (though often not in plain english!). You can find a selection at Yahoo.
The "birth" of the universe is thought to have been a point in time when all the matter and energy was condensed to a tiny point. This would have been incredibly hot and dense. The whole lot then exploded out at a speed which the human mind (well, mine anyway) simply cannot comprehend. The reasoning behind this is that everything in the universe is moving away from us, and the furthur things are, the faster they are moving (this does not mean that we are the center of the universe!).
To get to your question, there is nothing outside the universe for it to expand into. It neither has a "boundry" where it all stops, nore does it go on forever. In fact there is no "outside" the universe. I'm afraid it all gets a bit complicated here, and I must admit I don't really understand. The only half-way understandable explanation I've seen is in Stephen Hawking's "Brief History of Time", though even that made my brain hurt. In brief, what he say's is that the universe is effectivly curved round on itself (like the inside surface of a sphere, only in three dimensions, and not two!) so if you kept going far enough you would get back to where you started. Only you never could, because even if you travelled at the speed of light (itself impossible!) the universe would end before you got back. Right, I'm going to lie down in a darkened room....
Added by Marc Herant:
Chris is right, that's not an easy concept to be comfortable with. A balloon can expand in diameter without having an edge that pushes into something. Now, you'll say that the balloon is a two-dimensional surface embedded in three dimensional space and that it is expanding into that space. That's true in our experience, but mathematically, the balloon does not need to be embedded into anything, just like our three dimensional space does not need to be embedded in a higher dimensional space to exist. So, as a mathematical space, the universe does not need to expand into anything since it has no edge. Hmmm, I think I'll go lie down too.
Try the links in the MadSci Library for more information on Astronomy.