One of the most frequently asked questions at Ask an Astronomer--and one of the most infamously tough to answer--has to do with the expansion of the universe: what is at the end of the universe? Because if the universe is expanding, it must be expanding into something, right?
No. A number of previous pages (see "related questions") address this question, but I would like to stop for a minute to talk about why it is so tough for us to get our minds around this issue, and why answers that are technically correct can still be unsatisfying.
Trying to understand the expansion of the universe is a bit of a paradox--because really understanding it involves "giving up" on ever comprehending it in the same way that you comprehend your everyday physical world. I might tell you to picture the universe as an expanding raisin-cake or a four dimensional balloon (some favorite mental models), but it just isn't possible to grasp "the universe" in the same intuitive way that you understand, say, the shape of a ball in your hand.
That doesn't mean that it is impossible to understand the expansion of the universe, but it does mean that, for most of us, that understanding will be of a different quality than what we're used to--it will be grounded in mathematics rather than in physical experience.