For some systems with fast relaxation dynamics and being driven slowly, the systems respond to the driving force with sporadic relaxation events (the avalanches) instead of continuous re-adjustments. These relaxation events may lack a typical length scale and their sizes follow a power-law distribution thus the systems have diverging correlation lengths and become scale invariant (critical). As this behavior often observed without explicit tuning of the control parameters of the systems, we call it self-organized criticality (SOC).
SOC is one of the attempts made by physicists trying to understand the complexity of the nature outside laboratories, especially, the scaling phenomena.
What are scaling phenomena? Before answering that, let's point out a very lucky situation in the nature for scientists. That is, it's actually possible for human beings to develop science at all. There are two reasons for this. The first is that the nature actually follows a pattern of rules. The second is that these rules happen to separate into levels of details. The first reason is obvious as science is to study the rules of nature and, lucky us, their do exist.
The second one is not as obvious but it makes it possible for human being to grasp the rules of the nature. Let me give you an example, if the orbital motion of the planets depends on the detail structures of their constitution, it will be a lot harder for Kappler to come up with any laws of planetary motion. Because the separation in the levels of details, we are able to understand the rules of the nature in a step by step fashion. The levels of details are usually described by the length scales or, inversely, the energy scales, for example, atomic scale, planetary scale, high energy, low energy... These actually becomes a major way that we divide the field of physics into various sub fields and research can be carried out more or less independently in these sub fields.
However, there are also interesting phenomena where the details in all levels are equally important. And, it becomes very hard even to formulate the problems in commonly used language of physics. An important example is turbulence in fluid dynamics where large eddies break up into smaller ones which break up into even smaller ones. In these systems the physics of the phenomena we observed happen in all length scales. This makes it impossible for us to understand if we are to focus on a certain level of details. However, there are better situations that even when the degree of freedoms from all length scale are involved in contributing to the observable phenomena, things could actually look the alike in different length scales. This simplifies the situation and allows us to characterize a system just by the way it looks at one length scale and how it scales when the length scale is changed.
One tricky point here is that when we say the system looks alike in different length scales, it doesn't mean that the system is identical in different length scale. It only means that there is no typical features of the system that will allow you to tell what scale of the system you are looking at. For example, when you look at the coast line on a map, if you are not familiar with the area, it will be very hard to tell what's scale was used in making the map. This is what we mean by scale invariance.
In experimental system, to reach a critical state usually requires tuning
of control parameters to a critical point. Examples are the end point of liquid-gas
transition and Curie temperature of zero field magnetization. However, in
nature we often see a scaling behavior in a lot of places. As not every one
believe that there is something tuning certain parameter of the nature to
its critical point, some physicists come up with the idea that the certain
mechanism can self-organized into a critical state without explicit tuning
of a control parameter. And, this is so called Self-Organized Criticality.