“Tell me what it isn’t!”
The examples I gave, superconductivity and ferromagnetism, are example of phase transitions, which only happen when there are large number of components interacting. I wouldn’t call phenomena that can be explained by one or few components as emergent. So, I wouldn’t call a black hole as emergent. I wouldn’t call an electron and a proton making up a hydrogen atom as emergent. I wouldn’t call two hydrogen atoms and an oxygen atom forming a water molecule as emergent. But liquid that is formed by a large number of water molecules is an emergent phenomenon to me. Jagadul’s example of traffic jam is also a good one.
I disagreed Rip’s opinion that black holes etc. are examples of emergent phenomena, but other than that I don’t really see much disagreement about what emergence is in the comments here.
I like Zubon’s description “the whole is not predictable from the parts” and “No cell in your brain understands Chinese,..., but the system as a whole does.”
Why can it be useful? I can think of two possible reasons. There is a certain reductionist tendency (although I don’t think being reductionist per se is bad) to assume that we get better and better understanding of the system just by getting more and more detailed information about its components. But the concept of emergence reminds us that there are certain things that we only understand when we see the system as a whole. Another thing is that it allows us to see something common in very different systems studied in different disciplines. For example, there is similarity between spinglasses in solid-state physics and neural networks. Also, you can see so-called power law in many kinds of systems. (Although power law may be another example of a concept that is overused.)