Complexity refers to the study of complex systems, of which there is no uniformly accepted definition because, well, they are complex. Roughly speaking, one says that a system is complex if it consists of many interacting components (sub-units) and if it exhibits behaviour that is interesting but at the same time not an obvious consequence of the known interaction among the sub-units.
That sounds very vague, especially the use of words like "interesting" and "obvious", but it reflects an evolutionary perspective. For example, a hundred years ago one might have described the study of how a substance changes under heat (phase transitions) as a difficult and interesting problem that required one to deal with systems with a large number of interacting components (atoms). However by now very powerful tools, such as thermodynamics and statistical mechancis, have been developed to deal with such equilibrium systems leading to impressive quantitative agreement between theory and experiment. Though such systems are not commonly referred to as complex, they still provide valuable examples and concepts that have been used in complexity studies.
Current interest has shifted to dynamical systems that are (generally) out-of-equilibrium and thus highly non-linear. Such sytems actually form the bulk of natural phenomena but for which the theoretical tools are as yet poorly developed. Some examples of such complex systems or phenomena are: The economy, the stock-market, the weather, ant colonies, earthquakes, traffic jams, living organisms, ecosystems, turbulence, epidemics, the immune system, river networks, land-slides, zebra stripes, sea-shell patterns, and heartbeats.
There is no single "Theory of Complexity", and it is unlikely that there will ever be one. Rather one hopes that apparently different complex systems can be grouped according to some common features that they have, so that intuition and insight gained in studying one can be transferred to another. Thus one of the main aims of complexity studies is to develop concepts, principles and tools that allow one to describe features common to varied complex systems. This leads to exciting interdisciplinary studies because it turns out that ideas developed to handle complex systems in the physical sciences have relevance also for systems in the biological and social sciences, and vice versa!
What are some of the characteristics of complex systems? One often quoted concept is that of emergence, which refers to the appearance of laws, patterns or order through the cooperative effects of the sub-units of a complex system. Thus the emergent phenomena or laws are not an intrinsic property of the sub-units but rather something that is a property of the system as a whole. Simple examples are those of "temperature" and the "gas laws": At the individual microscopic level, none of those make any sense, but they are features of a large system. More sophisticated examples are of "intelligence" and "conciousness" - where do they come from ?
Sometimes one sees the phrase "the whole is more than the sum of its parts", as a definition of emergence. This again reflects the non-linearity of the system, whereby the output is not proportional to the input, small changes can give rise to large effects, and the non-obvious results that can be produced in a large system.
It is important to realise that the universe consists of many hierarchial levels of complexity linked to each other. Each level has its own emergent patterns and laws: As one goes down from galaxies, solar-systems, planets, ecosystems, organisms, organs, cells, and atoms to quarks, different effective laws emerge. However these laws would not be useful if there was not some degree of universality, that is, one hopes that at each level of complexity the same laws apply to varied systems rather than each following its own tune. It is the apparent universality of the laws of physics, for example, that makes the world comprehensible and gives us faith in its ultimate simplicity. For example, at the atomic level weird quantum mechanics rules, but larger systems are well described according to Newtonian laws, while engineers often use empirical rules, and so do the social scientists.
It appears that nature has chosen to be economical (or is that an illusion on our part?), so that the branching of trees or the air-passages in our lungs, the shape of coastlines or clouds, the form of cauliflower or a mountain range, can be described by fractal geometry: Such shapes are self-similar over a wide range of scales, thus implying scale-invariance, whose hallmark is the appearance of "power-laws". In an equilibrium system scale-invariance naturally appears at the critical point of a second-order phase transition, such as that between the liquid and vapour phases of water. However natural systems are out-of-equilibrium and the common appearance of fractals and power-laws in such systems is not as well understood. Self-organised criticality is the idea that many out-of-equilibrium systems naturally organise themselves, without external tuning or prodding, into a state which is at the threshold between complete disorder and complete order: That is, the system arranges itself into a critical state, and so displays scale-invariance and power-laws.
Living systems are the most complex examples one can think of and it is remarkable how such systems tend in their development towards greater order, organisation and complexity, in contrast to the arrow of time dictated by the Second Law of Thermodynamics. Of course there is no conflict as the increase in disorder and entropy required by the Second Law refers to closed equilibrium systems. Living systems are neither closed nor in equilibrium, but rather use an inflow of energy to drive processes that increase their order (thus decreasing their entropy), and dissipate heat and other waste products that lead to an overall increase in entropy of the universe. One can say that organisms are dissipative structures, and have a tendency towards self-organisation and pattern formation.
Ant-colonies are classic examples of self-organisation. Without a leader (the queen is actually an egg-laying machine) orchestrating everything, and without any of the ants having taken a course in engineering or social science, each ant seems to do its own thing, following a few simple rules that determines its interaction with its environment or its ant-mates. Yet, an incredibly complex and organised society emerges from such an interaction of the many ants. Ant-colonies display remarkable adaptation to changing circumstances, using both feedback mechanisms and parallel analysis of options. In recent years social and computer scientists have taken a keen interest in studying ant colony behaviour in order to help solve problems in their own fields.
Not all systems in nature appear organised or have some pattern to them. Indeed many seem disorderly or ruled by random events. However some of that randomness might only be on the surface. Chaos refers to the property of some non-linear dynamical systems whereby they become extremely sensitive to initial conditions and display long-term aperiodic behaviour that seems unpredictable. Though chaotic behaviour might appear essentially random, there is actually hidden order, apparent only in "phase-space" rather in ordinary space. Furthermore, many chaotic systems show universality in their approach to chaos, giving one some predictive power. Thus discovering that some random-like events are actually chaotic means one has uncovered a simple determinstic basis for the system and so enabled its understandability.
Often one encounters debates between reductionism and holism. Reductionists like to get right down to the bottom, meaning they are interested in the basic sub-units that make up the whole and believe that that is where all that is of interest lies, the whole itself being just a complicated and uninteresting consequence of the fundamental laws applied to a large system. In short, knowing the microscopic explains all to the extreme reductionist. Particle physicist are such, and in more recent times some molecular biologists involved with genomics are another example. While it is undoubtedly true that knowledge of the microcomponents of a system and the basic interactions among those is essential for us to progress, it is also a fact that such knowledge by itself is insufficient to predict all the diversity and novelty that can arise in a large system. (Take for example the task of predicting superconductivity from Schrodinger's equation - it is a problem that required much effort after the fact-one knew what to look for. Similarly knowing the whole genome code is not going to predict for us every feature of an organism or a society).
The problem of precisely deducing the whole (of a large system) from its parts is at least two-fold. Firstly it is a computational problem. Problems with a large number of degrees of freedom are too complicated for exact solutions, and for systems far from equilibrium, as complex systems are, they are also not solvable by the probabilistic averaging methods used for equilibrium systems. In recent years the growth of computer power at low cost has produced the first tool that allows large systems to be simulated or solved numerically. However this brings the second problem: Often one does not have full knowledge of the fundamental dynamics, or the initial conditions, or the problem is still too complicated to be handled directly even by computers.
Often what is required is some guesswork or intuition to reduce the actual problem to a simpler model which can then be tested on a computer. Computer simulations of simplified models let one test assumptions quickly, and when the results appear similar to the real world one can take it as plausible validity of the model and the assumptions. Qualitative similarities of course do not constitute a proof, because other models with different assumptions might give similar results, but at least the insight gained helps one to make further guesses and tests in a particular direction rather than being lost in a mess of detail. In fact one of the most important lessons computer simulations have taught us is that a large system with very simple local rules can give rise to collective behaviour of great complexity and variety, showing on the one hand that complex phenomena need not require complicated rules, but at the same time reminding us how difficult it is (without computers) to deduce the emergent behaviour from the sub-units and their interactions.
Thus studying the whole is as interesting as studying its parts, as novel structures and emergent laws arise at each level of complexity. The condensed matter physicist studying superconductivity is not going to be replaced by the string theorist, and neither is the ecologist going to be become obsolete because of the molecular geneticist. Explaining dynamic patterns, order and emergent laws of a complex system by understanding the organising principles among the sub-units is what might be called holism, the counterpoint to reductionism.