You can read a lot about the behaviour of emergence in the wikipedia page about it here. Here, I would like to lay it in simple terms and limited examples I can provide. Phenomenon of Emergence is as the literal meaning it suggests; Something is emerging. What gets emerged is unknown to us until we observe them and studies them. But the most intriguing part of it is not the emergent properties but the simplest reasons that gives birth to them. I can simply state the behaviour of emergence as - A physical property which results in exhibitance of new physical properties which was earlier unknown to the system, but they emerged now because of the collective working of two or more than two parts of the system.
Emergence of Water
Let me explain it in more simpler terms. Consider you have a molecule of water with you. Now study its physical properties. We can say that, it has two hydrogen atoms, one big oxygen atom in it. It has a bend structure and it can attract positive ions towards it. what more?
Now let's combine 1000 water molecules and assume they form a water drop. Now we can say that, it is in liquid state, it is transparent, It can freeze at zero degree celsius and melt at 100 degree celsius and there goes the properties. Where did these properties arise? Why did the collection of H20 molecules didn't flow? What if it was water vapour or ICE? From where did that new property arose? For now let's simplify it in the name of Hydrogen bondings. However, the real trick lies within the mysterious reasons of Phase Transitions. Yes, Phase transition is an emergent property which the H2O molecule never had but the water drop acquired. When single entities came together to work collectively, they exhibited new properties which were unknown to them or to any observer before. So, Emergence is the study of this new properties - How they arise, How they are formed, What made them to trigger and What are the consequences.
The above described example is a simple one which is related to the physical properties of a chemical molecule. There are various other examples which you might find amazing and would woah! when you think of them in this way.
Emergence of a swarm
Haven't you seen flies swarming around a light source? especially at the night time? they will be swarming around and yet they never hit each other. They aren't trained pilots to not collide, but yet they don't. Even our trained pilots would do shiver when it comes to flying as a group. How do the flies manage to exist as a swarm, even if they haven't ever studied on how to do that?
Emergence of Flocks
Have you seen birds flying in different shapes in the sky? Especially the migratory birds and mostly they form a V shape while they flock. How do you think they came to know how to fly like that? There is a lot of aerodynamic advantage for them to fly like that, rather than flying individually and randomly. You can understand why it is important by another example. Consider there is a party going on. Lots of people are gathered there. Now, if somebody screams SNAKEEE! what do you think people will do? They will randomly run, will hit each other and falls down, will kick the fallen ones and would run anyway. But, if it was a group of deers or buffalos in a forest and they saw a tiger or lion, they would too run like hell, but would never hit each other and falls down.That's how the emergence helps here. There are certain properties which a system would acquire when the individuals work together, which might not be seen when the individual was working alone; Even the individual is still not aware of it. It's still a mystery on how these system forms new traits when they works collectively. This is an active area of study which have only emerged :). You can browse more about it and do some research to expect the unexpected and utilise it.
