The “butterfly effect” is a term from nonlinear dynamics, which is a subdomain of physics. It occurs in systems that meet three requirements: the output is not always proportional to the input (“nonlinear”), the progression is dependent on time, but is a function of only the original state (“dynamic”), and randomness is not a factor (“deterministic”: if A, then B). When these three conditions are met, a small change to the initial conditions can lead to large changes in the results. The phrase was coined by the meteorologist Edward Lorenz, who was referring to a butterfly flapping its wings in the Amazon rainforest being able to influence the weather in Texas.
The phrase is generally used today when a seemingly insignificant cause can have a large effect. The quantum world does not exhibit deterministic behavior, however, so you can’t really speak of a butterfly effect there. There are, however, some parallels. Researchers are studying how quickly certain effects propagate in certain quantum systems. The most important effect here is decoherence, that is, the inevitable, but undesired disappearance of a fragile quantum state in favor of the normal world. To that end, you need to understand that quantum states unfortunately have the tendency to dissipate, primarily through interactions. This makes it difficult to build useful things like quantum computers, for example. The quantum state is ordered; decoherence spreads chaos, so this process is also called a “butterfly effect.”