When cosmologists try to explain the universe, they have a very interesting vocabulary. They use terms such as "entropy," "isotropic," "homogenous," and "finite but unbounded."
To try to explain the expansion of the universe, conceptually, is a difficult task. We have observational evidence of an expanding universe, but there is no "center" of that expansion, and no "edge" the the universe. No edge, yet it (might be) finite. So, how can something without an "edge" not be infinite?
Imagine the surface of a ball. A three dimensional sphere. On the surface, there is no "center." You also can draw a line on that surface, and go on forever without coming to any "end" of that ball. You will certainly draw over parts you have in the past, given a long enough line, but you will never hit a wall or fall off of the edge. Becuase the surface is "finite but unbounded."
That's an example of a two dimensional surface (surface of the ball) in three physical dimensions (the "sphere" of the ball).
When you ask, "what is the universe expanding into" (a very good and very common question) the answer is, in general, either "itself" or, it is expanding into a fourth physical dimention.
Go back to that "ball" and imagine you were able to inflate it. The distance between any two points on that ball is getting larger, but the ball is still finite. And it is not expanding into anyhing two dimensionally, but rather it is expanding in 3-D space. But the "2-D" surface, appears to be expanding.
This is the closest I can come to explaining the expansion of the universe. Take that analogy, and think of it as 3-D space expanding into a 4th physical dimension.
"What will happen when it stops expanding?"
I've already typed up a lot, but there are a few theories as to what the fate of the universe may be. I personally subscribe to "The Big Rip" but haven't conclusively ruled any out. I will leave the wikipedia articles on several in my sources.