This is the part of chaos about which most people hear: it has been mentioned in Jurassic Park, Dilbert comics, and even in a recent Venus Williams DMX endorsement commercial. This is the idea that "a butterfly stirring the air today in [Beijing] can transform storm systems next month in New York"1 This statement, although somewhat weird, actually provides and interesting explanation of one of the most fundamental behaviors of chaotic systems: sensitivity.
Not that kind of sensitivity. Sensitivity is basically the idea that small differences in initial conditions have drastically different effects on the system over the long run: basically, it is a generalized rule for the Butterfly Effect.
This kind of sensitivity can be seen in weather systems where it was first discovered by Edward Lorenz.
Initially, things were looking pretty good. Lorenz then saw the pattern start to repeat itself. It was what happened next that was startling. Slowly, the path of his original weather pattern split from the weather pattern of his second run. After a while, the patterns became completely different.
You see, Lorenz had his computer printouts stop after the fourth decimal place. In reality, however, the computers held data to several more. During his initial run, Lorenz had unrounded values. By using the rounded values, Lorenz had seen the sensivity on initial conditions that marks system. I guess rounding to the fourth decimal place may have represented that Beijing butterfly flapping its wings!