Emergentism on the other hand says that beyond a certain level of complexity something new emerges which has to be explained in its own terms and cannot be explained as the interplay of less complex systems. In other words it says you have to explain psychological processes in psychological terms, not as electrical or biochemical activity in the brain. You have to explain psychological phenomena in psychological terms rather than biological ones, biological phenomena in biological terms rather than chemical ones, etc.
No idea about supervenience. I agree with Bingley about the reductionism. Reductionism is in opposition to holism. Reductionism attempts to explain something in terms of it's constituent parts. It works well when one is trying to explain phenomena in which one actually does know what the parts are, how each of them functions, and what the interrelationships are between the components. Think of this in terms of trying to explain things causally.
Holism (not the quack medicine, but the philosphical) is an attempt to explain a thing as it relates as an individual entity to its environment. An example of holism might be Skinnerian psychology - but it's not a particularly good example. (I'm afraid I'm in a rush at the moment and don't have time to give this the attention it deserves.) Holism works when either we don't know what all the important components are, we don't understand them well, or their interrelationships are intractable to current analyses.
I don't think this differs too much from what Bingley said. However, where I don't think we agree is in the meaning of emergence. Emergence - at least in a computer sciece / complexity theory context - is what happens when you get an apparently complicated (or complex) or emergent behavior from very simple rules.
The paradigmatic example is the flocking of birds which is based on three very simple rules.
http://www.mgtaylor.com/mgtaylor/jotm/spring97/flock.htm
http://www.red3d.com/cwr/boids/
More recently there was an article about a guy at MIT who built a set of robotic bees that worked together. (Not having read the original paper, I've no idea whether the experimenter has observed emergent behaviors in his cyber-bug society.)
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