To put it shortly, a quantum simulator is a quantum system, whose set of governing equations makes it behave exactly like another quantum system, which may be unreachable experimentally.
For example, the "zitterbewegung" motion of electrons that was predicted by the Dirac equation has so far been impossible to measure experimentally due to the extremely small values of the motion. However, a couple of years ago, the Innsbruck group of ion trapping showed that they could tune their ion trap experiment, by changing the light and phonon interactions of the ions in just the right way, such that they could rearrange the governing equations for their trapped ion, making the equation look exactly like the Dirac equation. Now all they have to do is let the trapped ion system evolvel naturally and they will observe the ion doing the same thing that the electron governed by the Dirac equation would have done.
Also note that the difference between a quantum simulator and a quantum computer is that a computer is very general and can be programmed to solve many different things, whereas a simulator is very specific and can only solve that which has the same equation as itself.
