PBS/NOVA has a great, short, easy to understand video on emergence. It describes the prominent status of “emergent complexity” research in the sciences, largely focusing on applications in natural sciences and biological examples of herd behavior in particular.
Here are a couple of highlights from the video:

• Most of science’s unsolved mysteries are counterintuitive emergent complexity problems
• Emergent complexity (or emergence) arises from many agents following simple rulesets and no leader
• Emergence requires many agents with many more interconnections
• Flocks of birds, schools of fish, and human crowd dynamics “moving as one” are all emergent results of agents exhibiting rule-based behavior.
• Craig Reynolds demonstrated birds flock spontaneously by programming a simple ruleset of bird behavior into a computer simulation.  (See it here)
• Learning in computers and AI is a form of emergence – computers can be programmed the rules of checkers, a strategy to favor moves leading to victory, and adjust their moves given the moves of human players, and can eventually beat human players.
• The number of connections can be an upper limit to the extent of emergence – human brains can outperform computers since each neuron (of billions) is connected to 10,000 other neurons, whereas any computer transistor connects to only 10 others.  Consciousness may be an emergent result of these connections.
• Emergence may explain abiogenesis (the origin of life) – Bob Hazen’s experiment with pyruvic acid demonstrates the possibility of biological life emerging from simple molecules spontaneously ordering themselves into more complex structures given substantial heat and pressure.