This talk has three parts: 1. I discuss the phenomenon of understanding – our ordinary experience of understanding the objects and events in our environment. Normally we do not pay attention to the understanding process itself, but just to what is understood (e.g. I understand the ordinary things you say, but do not enquire how it is that I understand them); here we focus upon the process. I argue that understanding holds an important key to the nature of human cognition—our ability to think and reason. 2. Next I examine the process of self-organization – the process whereby a type of general order arises from local interactions between parts of an originally chaotic system. Self-organization is so-called because the order is not controlled by any agent external to the system. Many familiar phenomena are self-organized, from rush-hour traffic patterns to ant hills, as well as many organic processes within our bodies. 3. Finally, I attempt to show that understanding is a self-organizing process. In considering cognitive functions in the brain, I take a top-down rather than a bottom-up approach. A top-down approach starts with a general system, in this case our conscious awareness of understanding, and breaks it down into (sometimes unconscious) subsystems. I argue that the sort of understanding we are familiar with is possible only through the self-organized subsystems of our ordinary understanding of our situation and environment.