The butterfly effect is an idea that says that a small change can make much bigger changes happen; that one small incident can have a big impact in the future. The idea started from weather prediction. Now the term is used for many things; some are scientific, and some are not.
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.