What is the universe expanding inside of?

The universe is all that there is, it doesn't need something to expand into, it expands within itself.

You know I have always wondered that as well.
Cap, your answer doesn't really explain anything.
Now is this because our understanding of what is "outside" the universe so scant that we truly have no idea ?

I always thought of the universe expanding like a balloon being blown up with air, but obviously not, or maybe, who knows!

Okay I shan't get into semantics here. Universe means "all turned into one". It means "all that there is that exists, including time and space". If we discover a point outside the universe, it immediately becomes inside our universe, because that's what universe means.

Let's ignore this and assume that there can be something outside the universe that isn't included in our definition of universe... does it need to be there for expansion?

The ballon analogy you use only goes so far, curly. It illustrates how a 2D surface can expand from within all points on that space. Analogously, our universe expands in 4D space. If you look at the balloon, the 2D space doesn't expand into anything in it's 2 dimensional space, it gets bigger without expanding into anything. In this same way, our 4D universe expands without expanding into anything.

Now I can see what you're going to say "the balloon expands into unoccupied space in the third dimension". You'll have to accept my word on this, the third dimension is not needed to mathematically expand a 2D surface. It does not need a space to expand into. The 4D version of this, without need for a higher dimensional space to expand into, is our best guess at how universal expansion works.

It really doesn't need any space to expand into... You need to remember that the surface of the balloon, like our universe, is unbounded.

Please ask further if I've failed to make it clear yet (I'm sure that's entirely possible), and I will do my best to keep explaining :)

I agree with capuchin.
In addition, discussions of things outside the universe have no value. As capuchin implied, the universe is defined as all the things which we can observe in some way. Anything outside the universe can not have any influence on us and therefore is irrelevant. Any discussion on such a topic may be fun but will ultimately result in pure speculation. I could say that the universe is expanding inside of anything I want... but it doesn't matter because I could never be proven correct or incorrect.
Besides, I could also argue that the universe isn't expanding, the resolution is just increasing. :)

Curleyben illustrates the biggest common problem with the balloon analogy of the expanding universe. As Capuchin explained, the expanding universe can be described as the surface of an inflating balloon. The area inside the balloon and outside of the balloon do not exist as far as the analogy is concerned.

Time and space began with the big bang. What exists outside of time and space? No, God does not count as an answer.

It could be more time and space, it could be a whole universe dominated with clowns, it could be flatland (http://en.wikisource.org/wiki/Flatland), it could be nothing. Whatever it is, it is currently both unobservable and undetectable. So the discussion is not particularly useful until we have evidence that infers it's existence.