The simulation of a quantum computation (some people choose to use the term 'emulation' in this context to disambiguate from a different type of simulation) is when one tries to recreate the calculation that you want a quantum computer to perform, but on a classical computer.
When you're simulating a particular algorithm, there are many different problem instances with problem sizes $n$ (this is usually the number of bits required to specify the problem instance). We say a problem is hard if the time that it takes to run grows quicker than any polynomial in $n$.
Now, it must be emphasised that we don't know that simulating a quantum computer is hard. It's just that we don't know how to do it. And we've tried quite hard. For example, we believe that quantum computers can perform some classical computations that are hard at least as strongly as we believe that there's no efficient classical algorithm for factoring large composite integers (because there's a quantum algorithm that achieves that in polynomial time).
If we could prove that classical computers can simulate quantum computers, then there wouldn't be nearly so much interest in building a quantum computer. That said, simulation is a polynomial overhead equivalence. Quantum computers could still be much faster, which might be desirable in some contexts.
So, your question effectively boils down to "where do quantum computers get their power from"? Variations on this theme have been asked a number of times already on this site. The way that I like to think about it is to recall that classical computers, no matter how complex, are built out of the same fundamental set of gates (indeed, one gate such as NAND is sufficient). If somebody suddenly comes along with an extra gate that cannot be built out of the existing gates, it suddenly gives you the potential to use this gate to improve existing algorithms. Sometimes it'll help, sometimes it won't.
I would just like to point out that one aspect which is not the source of power is the exponential state space. Probabilistic classical computations also have an exponentially large state space, and yet we can still perform them. (Of course, the difference is about how we deal with probabilities. Quantum probabilities can interfere, which means that we have to keep all the paths "alive" as we simulate, rather than just sampling individual paths. But this is a much more subtle issue.)