A commonly encountered element in most timelines. It describes the secondary, minor effects of every Divergence that add up over time. While every Divergence has very direct consequences that can be detailed and explained in the timeline itself, it of course also, realistically, changes the lives of thousands or millions of people in the timeline in ways too small to describe in detail. However, those effects have effects themselves, etc. etc., which leads to a change in the timeline's world not covered by the direct timeline itself. As those changes cannot be explained in detail, they are summarised as “Butterfly Effect”, to which (from the author's and reader's point of view) random elements in the timeline are attributed.
One important factor in that is that as soon as the Butterfly effect takes hold (varying on approach, see below), people now born will not be the same person as OTL (though they may have the same name), because even if the parents should still meet, the chance of the same sperm meeting the same egg is infinitesimal. This is also one of the points of controversy between the approaches, again see below.
The name derives from a famous example used in chaos theory: The idea that a butterfly flapping its wings in Europe can later cause a devasting storm in China - as it will create a tiny wind current, which will go on to upset larger and larger wind movements all over the world. Six months later, the timeline where the butterfly flapped its wings has stormy weather, while the one where it didn't is sunny. Of course, the flap may also prevent a storm that otherwise would have happened…
While there is a broad consensus on the forum that the butterfly effect must be taken into account to cover all the secondary effects, there is some controvery about the application of it. The most 'fundamentalist' approach says that absolutely anything that happens after the PoD has the potential to go differently to OTL.
The other, more moderate approach to the butterfly effect is to assume that the Butterfly Effect spreads outwards from the origin of the Divergence, reaching more far away lands only after some time, time the secondary effects need to build up and reach those lands. Even then, the moderate approach will tell you the butterfly effect only affects the “context system” of the Divergence's region. For example, if the PoD happens in medieval Peru, then it may take some decades for the butterfly effect to manifest in Mesoamerica, and even centuries for it to reach outlying regions, while European history will not be affected until 1492.
The third approach, to disregard the Butterfly Effect more or less exactly, or to have it affect or not affect the world at the author's arbitrary will, is as said pretty much discredited on the Forum, though very popular in professional literature.
Reasons for such a liberal, or at least a moderate approach can be found in literature and writing stories: Henry VIII on an airship is more exciting than English King You've Never Heard Of on an airship. Alternate History in stories works much better if the reader can spot familiar themes - people, nations or events. That is also why the liberal approach is so well liked in popular literature. And of course it is easier to write a timeline if you do not have to worry how history goes in those parts you do not focus on, as you can simply assumes it goes as per OTL. On the other hand, as the namesgiving example of chaos theory tells, changes anywhere on the world can have effects anywhere else, so the fundamentalist approach, or at least a strict moderate approach can be seen as more realistic, hence timelines aiming for that usually employ such an approach. Also, it allows to shift focus to different regions of the world, so timelines with the aim of completist coverage of the world also tend towards that.
Fundamentalist approach: Decades of Darkness
Moderate approach: The Guns of the Tawantinsuya
The Citroen DS Incident, the antithesis of the Butterfly Effect.
General Theory of Alternate History - A discussion by various AH.commers further ruminating on the topic.
What to do about butterflies ? - Discussion on handling the butterfly effect narratively (via butterfly nets or similar solutions), in order to avoid things becoming too complex or too ambitious to cover by one author in a reasonable amount of time.