Let me take malawi_glenn's explanation one step further, if I may.
Imagine you are watching a movie of a ball that's spinning clockwise bounce off a wall and go in some direction. Now imagine watching another movie of a similar event, but now the ball is spinning counterclockwise, and it bounces off in the opposite direction (to conserve angular momentum).
QUESTION: How do you know if the second movie you saw was a "real event", or just the first movie, with the camera recording through a mirror?
There's no way to know: both scenarios would give you exactly the same movie!
**UNLESS** parity is not conserved. For example, if the "wall" the ball bounced off of was a W-boson, the movie with the clockwise-rotating ball would NEVER happen in the real world, so now you would KNOW for sure that it was not a "real event".
Of course, my example is a little silly since clearly quantum helicity and spinning balls are not the same thing, but this is just to give you an intuition, not to be taken too seriously. The point is that if parity was conserved, the mirror world and the "real" world would both exist. If parity was violated, then only one of these worlds would exist. The other would only be an illusion.
Hope that helps.