Dynamical systems are called chaotic when it is impossible to predict what the system will be like at some future time. A mathematically precise definition is given in the next section.
To understand chaos, we first take an example of a non-chaotic system: a cannon shooting cannonballs. If we shoot a cannonball of certain weight, with a certain amount of gun powder, and at a certain angle, it would land at a specific distance. If we then shoot a cannonball of slightly different weight, slightly different amount of gun powder, and slightly different angle, then we would expect it to land at a slightly different distance than the first cannonball.
On the other hand, weather would be an example of a chaotic system. If we have some initial state of certain humidity, temperature, and wind speed, then a certain weather state would be produced two weeks later. If we started with some initial state of very slightly different values of humidity, temperature, and wind speed, the weather state produced two weeks later would be completely different from the the first. While this seems hard to believe, try to remember the last time meteorologists predicted a hurricane or tornado two weeks beforehand. They cannot because weather is a chaotic system.
It is important to note that chaotic systems are still deterministic. That means if it were possible to have infinitely precise data about all of the variables involved and how they changed in time, and we had an infinitely powerful computer, all future outcomes could be calculated precisely.
A weather forecaster can only ever give you an estimate of who likely it is to rain--they simply cannot tell you how many rain drops will fall on your head. This example also illustrates another common property of chaotic systems: while we can't say anything exact about where the system will be in the future, we can make general statements about how likely certain outcomes are.
While not all chaotic systems can be analyzed this way, describing probabilities can be a powerful tool for analyzing many systems.
A More Mathematical Definition
Ideas for the Future
Interactive animations of the cannon/cannonball system and of a chaotic system.