The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory. Small variations of the initial condition of a dynamical system may produce large variations in the long term behavior of the system. This is sometimes presented as esoteric behavior, but can be exhibited by very simple systems: for example, a ball placed at the crest of a hill might roll into any of several valleys depending on slight differences in initial position.
sensitive dependence on initial conditions in chaos theory is commonly presented in this. "A butterfly flaps its wings in Japan which causes tiny changes in the atmosphere, which ultimately cause a tornado to appear half way across the world. This is where the term "The Butterfly effect" comes from.