When a property of a system cannot be traced back to any of the individual parts in the system, it is called an emergent property. Your personality is an emergent property of your brain. It cannot be traced back to individual neurons. Likewise, fluidity is an emergent property of water, and culture is an emergent property of a group of people.
Three aspects are important for a property to be emergent: First of all, supervenienceis the observation that the property will no longer exist if you take away the individual parts of the system. For example, your personality disappears when I remove your neurons. (Relax, I won't try and prove it.) Furthermore, the property should not be an aggregate, meaning that it is not simply the result of adding up the properties of the individual parts. For example, a single water molecule has no fluidity. So you cannot simply add up the fluidities of a billion individual molecules to determine the fluidity of water. Finally, there must be downward causality, meaning that the emergent property should influence the behavior of the individual parts. For example, the culture of a group of people influences the behavior of its members. In short, emergent properties are global (to the system), irreducible, and noticeable (to the parts).
The boundaries of the sciences rely on emergent levels. Physics resolves into chemistry, which resolves into biology, which resolves into psychology, which resolves into economics. And each science works with the emergent properties generated by previous levels. It also means that each level gives rise to new laws and new insights. Psychology is more than just applied biology, biology is more than just applied chemistry, and chemistry is more than just applied physics. This is why greedy reductionismdoesn't work. You cannot explain forgetting your spouse's birthday in terms of faulty atoms or string theory.
Throughout literature there has been some confusion (and disagreements) about self-organization and emergence. Some scientists have defined one in terms of the other, while others claim they are separate concepts. I tend to agree with Peter Corning in that there may be self-organizing systems without emergent properties, and emergent properties in systems that are human-created instead of self-organizing. But it is just a matter of definition. On this blog I choose to use the term emergence for “organized ‘wholes’ composed of functionally distinct ‘parts’ that produce irreducible combined effects”. Even though this blog itself is not a self-organizing system, the impression it will make on you will definitely be an emergent property: the impression is global to blog, irreducible to the individual words, but quite noticeable if you choose to iniate a DDoS attack at my blog after reading.