A tricky part of this question is the wording: "the universe," by definition, is all there is! To speak of the universe expanding into something would mean that there was something bigger, which we ought to have called "the universe" in the first place.
Perhaps the easiest way to see what is meant by an expanding universe is to imagine what life would be like for two-dimensional ants living on the surface of an expanding spherical balloon. They can crawl around, but being unable to fly, or to penetrate the balloon's surface, they live in what is essentially a two-dimensional world. For the ants, provided nothing disturbs them from outside, the universe is the surface of the balloon--that's all there is! Being confined to the surface of the balloon, there is no way for the ants to discover anything at all about what we would term up and down.
This two-dimensional universe is finite (the area is finite), but nowhere will the ants find a boundary or an edge. Here you have to ignore the rubber neck of the balloon and the person blowing it up; think of a balloon sealed smoothly into a spherical shape hovering in a tank in which the air pressure could be lowered to make the balloon expand.
Now as the balloon expands, the ants see each other getting farther and farther apart. Each sees the same thing: all its neighbors are moving away. The ants live in, what for them, is an expanding universe, with no physical edge. If an ant walks quickly enough, she could conceivably get all the way around the balloon and return to her starting point without encountering an edge.
You may object at this point and claim that we see the balloon expanding into the surrounding space. But we have access to an extra dimension in which to move--the one that would correspond to up and down for the ants, were they able to move in those directions. As far as the ants are concerned, they can learn everything they want to know about their world by making measurements on the balloon's surface, with no reference to the surrounding space. The surrounding space is inaccessible to them.
Experience in physics has taught us that when we find a concept that is spurious--in the sense that it leads to no predictable effects-- we do better just to assume it's not there. In other words, the ants would do well not to talk about their space expanding into something that they can't measure. Nothing is lost and there is a substantial gain in simplicity.
To make an analogy between the ant's situation and our own, you have to imagine space expanding in all directions. Everybody in the universe sees everything rushing away from everything else; but the universe need have no physical edges, and there is no need to describe it as expanding into anything; it can just expand.
Robert Nichol, an assistant professor in astrophysics at Carnegie Mellon University in Pittsburgh, provides some additional details.
The first and quickest answer is to ask another question: does the surface of a balloon have an edge? Clearly, no. Even so, as you blow up a balloon, the surface expands. The analogy also satisfies Hubble's Law in that were you to write dots on the balloon's surface, they would recede from each other more quickly as their distance from each other increased.
The balloon analogy is often used, but it breaks down when people ask: "What about inside the balloon?" After all, we are observing the balloon in three dimensions. An ant on the surface of the balloon does not know about up or down, and so to the ant, the inside of the balloon doesn't matter. Just the same way, the universe is expanding in space-time, or four dimensions at once (this revelation was Einstein's great leap). So just as the ant on a balloon lives in a three-dimensional universe but only "sees" two dimensions, we live in a four-dimensional universe but really only experience three dimensions.
To ask what is out beyond the edge is meaningless in space-time. First, there may not be an edge--this depends on the exact geometry of the universe. And second, there is expansion in space-time, so before the "big bang'' there was no space or time. So we are not expanding into something, because that something never existed.
