``Complex Systems'' has become a buzzword applied to a multitude of systems including the climate, social systems, economic markets, and engineered systems. The common characteristic is the interaction of many components that gives rise to ``emergent'' behavior that is not merely a large-scale version of the behavior of an individual component.
Emergence comes in different flavors, ranging from rather trivial to hightly sophisticated. Trivial kinds of emergence are aggregations that physicists often take for granted, such as the fact that one can define ``temperature'' and ``pressure'' and ``density'' in a meaningful way when all that one ``really'' has is a lot of gas molecule banging into each other.
One level above this are simple collective modes of behavior, such as waves: it does not make sense to speak of sound waves when one is looking at one gas molecule. Interestingly, these collective modes take on a life of their own in Quantum Mechanics, where sound waves, charge waves, spin waves and so on follow many of the same rules as ``elementary particles'' do. Consequently, they are dubbed ``quasiparticles'' and treated as such, and one can speak of the collision of an electron with a phonon (a sound wave quasiparticle). This opens up interesting questions about the status of ``real'' elementary particles, which for all we know may be collective modes of some underlying medium.
Another level is the emergence of phase transitions and other ``surprising'' phenomena: a system shows qualitatively different behavior (liquid/solid, random/coherent, steady/turbulent) depending on the values of some control parameter such as temperature or flow velocity. Some systems seem to adjust themselves to be right at the boundary between two phases; this has been called ``self-organized criticality'' (Per Bak), and may (or may not) apply to the weather, markets, earthquakes and other phenomena.
Even beyond this is a level of emergence where the structure of interactions between all components is what gives rise to a property that may have little to do with individual components. The paradigmatic example is protein folding, where the chemical interactions between amino acids cause the protein chain to fold into a three-dimensional configuration; it is the physical shape that then gives rise to its functionality, much like a key can be made out of pretty much any material as long as it has the required shape. Likewise, it is the arrangements of bricks that gives rise to the wall-ness of a wall, not anything in the concrete or clay itself.
Truly complex systems typically show different levels of emergence, with interactions between events on different scales that sometimes propagate to other scales, sometimes not. As a hypothetical example, the functioning of an organization of many thousands of people may depend on the skills of its president. The functioning of this president depends on the workings of his/her organs, which can be knocked out by a particular virus. This pathogen escaped the immune system because of a mutation in one of its genetic base pairs, which rendered it invisible to present antibodies. Organ fails, president dies, company collapses, disgruntled workers start revolution, world war breaks out.
The problem is that causal chains in complex systems are never as clean as I made them in my example. Many factors influence everything that happens, in indirect and intractable ways. Thus many of our traditional questions like ``did it happen because of A or B?'' really become meaningless. This is a problem for science - we always ask questions like ``all other things being equal, what difference does changing X make?'', when ``all other things being equal'' is just not an option in reality. It is also a problem for politics - one sees that a certain situation presents a problem, and one would like to change it, but any action one takes will trigger other reactions that are hard (often impossible) to predict. Put briefly, a complex system does not neatly decompose into independent subsystems that can be changed at will.
This also influences the way scientists approach complex systems. A complete understanding is often impossible; thus, one often aims for partial answers first, along the lines of ``what is the simplest system that reproduces a certain behavior that I am interested in''. Often it turns out that there are many mechanisms that can give rise to the same behavior, which may be in play at the same time, or at different times in different combinations. Telling which is which in the real world is often impossible, but not always necessary.