Is it true that a butterfly flapping its wings can result in a tornado in a distant location?
The butterfly is a colourful illustration of Chaos Theory, and the word butterfly came from the diagram of the state space (see below).

A system that is chaotic is extremely sensitive on its initial value.  In principle, if you know exactly how the state of the universe is now, you could calculate how it develops (but due to other reasons, it is theoretically impossible to know the state exactly — but that's not the main point here).  The issue with a chaotic system is that a very small change in the initial state can cause a completely different outcome in the system (given enough time).

So, suppose that we take the entire atmosphere and calculate the weather happening for the next 20 days; suppose for the moment that we actually do know every bit.  Now, we repeat the calculation, but with one tiny tiny bit that is different; such as a butterfly flapping its wings.  As the nature of a chaotic system is such that a very small change in the initial value can cause a very large change in the final state, the difference between these two initial systems may be that one gets a tornado, and the other doesn't.

Is this to say that the butterfly flapping its wings results in a tornado?  No, not really.  It's just a matter of saying, but not really accurate.

Many systems are chaotic:


Try to drop a leave from a tree; it will never fall the same way twice.
Hang a pendulum below another pendulum and track its motion:


 
(Figure from Wikipedia)


Or try to help your boyfriend in what must be one of the loveliest illustrations of Chaos Theory ever.  Suppose you are running to catch the bus.  You keep sight of a butterfly, which delays you by a split second.  This split second causes you to miss the bus, which later crashes into a ravine, killing everyone on board.  Later in life, you go on to be a major political dictator starting World War III (Note: this is not the plot of the linked movie, but my own morbid reinterpretation).


Tell me, did this butterfly cause World War III?

Not really.


(Figure from Wikipedia)