There are a few definitions out there. I like this one:
“The arising of novel and coherent structures, patterns and properties during the process of self-organization in complex systems” — Jeffrey Goldstein
The important thing is the collective:
Somewhere there's a critical mass that makes something new. But the boundary between too few and enough is fuzzy....
Here's a very incomplete list:
Here's an example of a "random" process. Click any three points in the canvas below. Then the app will plot a sequence of points as follows: The first point will be one of your original three. Each subsequent point will be a point halfway between the previous point and a randomly selected point from your original three. Do you think you will end up with a solidly filled in triangle? Try it!
Here's a question: Do you think the phenomena of quantum locking is emergent? Here's the original video:
Also see a more detailed video of the effect and a TED talk on the subject.
Studying emergence is very beneficial, both intellectually, and perhaps spiritually. Why?
If you like TED talks (an no one says you have to), here's one that focuses on synchronization:
David Chalmers has some ideas on emergence in this video:
Here is an essay and a book chapter touching on emergence and the notions of the spiritual and sacred: What do you think about them?
Reductionism has been successful in science, but Schrodinger's example of isomers in What Is Life? among many other examples shows emergence to be a powerful idea.
Sometimes you will see papers complaining about emergentists excesses, like this one.
Also it has argued by some that one should be careful in accepting emergent explanations for some phenomena because that may stifle scientific inquiry.
Can emergence be explained reductionistically?