What is Chaos theory?

Through history, until relatively recently, there has been at least an implicit feeling that once we knew enough about the world, we could predict anything. There was debate about whether this could be pushed to the logical extreme where if we measured all inputs, could we, say, absolutely know a person's thoughts because we would know the movements of the atoms and impulses in a person's brains?

But, forget mind-reading... I'd like to know if it will be sunny or rainy in ten days when I have a picnic planned? Well, it turns out that the forces that create weather are so "chaotic" that very slight changes can snowball into significantly different outcomes (rain vs. sunshine) over time. The shorthand is the “Butterfly effect:' something about a butterfly flapping it's wings in China means rain in Iowa.

Which leads to the conclusion that, because we can never hope to actually measure weather data with enough accuracy in the quantities that we would need, we will never be able to determine weather precisely at a date far in the future (e.g., ten days).

Chaotic systems are all around us. Drop five sheets of paper in exactly the same way in a still room and observe their different decent paths.

So, getting to the question: Chaos theory studies systems that are chaotic, like weather or air turbulence. These things were studied before, but it was a considerably philosophical and scientific leap forward to realize that they are all manifestations of the same class of phenomena and that many times very different chaotic systems can be described (stochastically) using the same mathematics. The math itself has lead to breakthroughs in other areas. As a trivial example, in the random generation of things like branches and shorelines in computerized graphics.