Chaos theory is a part of mathematics. It looks at certain systems that are very sensitive. A very small change may make the system behave completely differently.
Very small changes in the starting position of a chaotic system make a big difference after a while. This is why even large computers cannot tell the weather for more than a few days in the future. Even if the weather was perfectly measured, a small change or error will make the prediction completely wrong. Since even a butterfly’s wings make enough wind to change weather, a chaotic system is sometimes called the "butterfly effect". No computer knows enough to tell how the small wind will change the weather.
Some systems (like weather) might appear random at first look, but chaos theory says that these kinds of systems or patterns may not be. If people pay close enough attention to what is really going on, they might notice the chaotic patterns.
The main idea of chaos theory is that a minor difference at the start of a process can make a major change in it as time progresses. Quantum chaos theory is a new idea in the study of chaos theory. It deals with quantum physics.
The term butterfly effect comes from an analogy where a butterfly flaps its wings in Chicago and a tornado occurs in Tokyo. Weather is hard to forecast, because small changes like this can have a big impact on the weather. Therefore, this example is possible.
Besides weather, this term can be used in other areas where a small change can have a large impact. For example, it's one reason why many people believe time travel to be dangerous. If a small change was made to history, that could have a very bad impact later on. The butterfly effect serves as a metaphor for life in a chaotic world.
The Butterfly Effect shows that we cannot predict the future or control powerful complex systems.
In popular culture, the term “butterfly effect” is almost always misused. It has become synonymous with “leverage”—the idea of a small thing that has a big impact, with the implication that, like a lever, it can be manipulated to a desired end. This misses the point of Lorenz’s insight. The reality is that small things in a complex system may have no effect or a massive one, and it is virtually impossible to know which will turn out to be the case.