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A self organizing system is normally a multi-agent system where every agent does a specific action and a structure or pattern appears at the system level, without any specific global action by every agent.
Every agent will perform a specific action, usually each will have the same set of actions, and by doing so, a special global action is performed. This is called the emergent property of the self organizing system.
Let''s give a simple example: self organizing maps. Given a map of pixels, each with a color raging 1-255, which is initially quite indefinite in a shape, can be reorganized by a self organizing agents in order to find parts of the map which have a specific color dominance.
The Kohonen algorithm works in the following way: every agent will randomly pick a certain radius area of the map, and then choose the winning pixel, which is the pixel with the color closest to the area average. Then all the pixels are updated with a certain fixed percentage in order to get closer to the winning pixel.
The radius decreases in time, allowing smoother modifications.
In the end the system will emerge into an organized maps, with colored areas according to the most dominant color of that specific area.
We can observe that after agents that initially did a simple job like modifying colors in a slight matter, the system emerges into a different system having a specific pattern/structure/property, which is organization, in our case delimiting portions of the map having dominant colors.