The Banach-Tarski paradox says that you can cut up a ball into pieces, then rejoin the pieces into two balls that look the same as the original ball. Yep, that's why it's called a paradox.
Anyway, it assumes we start with a mathematical ball with infinite number of points in it. With that, the result seems plausible since you can cut an infinite set into multiple infinite sets, each of which contains elements that can be mapped one-to-one to the elements of the original set (e.g. the set of integers can be cut up into the sets of odd and even integers.) However, I never understood how the pieces of the original ball can be reassembled into two balls.
I tried to read the wikipedia entry on Banach-Tarski paradox before, but I still didn't understand. However, today someone pointed me to a web comic that explains how to cut up the ball into 5 pieces (and not 4 pieces) and reassemble them into two balls. I think it's the clearest explanation for non-mathematicians.