Emergence is the idea that complex systems adapt and create patterns and phenomena that go beyond their base, constituent parts. A classic example of emergence is a murmuration of starlings.1
Suppose you were going to write a computer algorithm to replicate flocking birds. How would you program it? Each bird needs to 1) align its direction with other nearby birds (positive loop, alignment creates more alignment), 2) move towards other birds, and 3) move away from other birds if too close (negative loops, proximity self-correction; finding a dynamic balance between too close and too far).2
Emergence is the complex balance between birds haphazardly crashing into each other and one lonely bird flying along all by itself. An emergent system is decentralized, so it is adaptive. For example, introduce a predator to a flock of starlings or a school of fist. Think of a hawk or a shark as a source of chaos—and watch how the system adapts.
A school or flock requires a dense network of interactions. A handful of birds or fish are inadequate. You need thousands of them.