A difficult-to-define concept. One things is clear though: it is not the same as complication; the human body with its interlocking systems and rhythms is complex while a bad piece of theatre is complicated because it does not cohere. Intuitively, complexity is usually greatest in systems whose components form intricate patterns and lowest when a system is either highly regular or completely disordered. Complexity is also applied to systems in which the outcome of some process is difficult to predict from its initial state. There are two broad classes of complexity: static complexity as in the intuitive notion of complexity, and dynamical complexity in terms of how much computational effort is required to describe the information content of a state of a system. A system can have low static complexity (or rather simple structures) and yet display complex dynamical behavior.
See Chaos, Complex system, Dissipative system, Dynamical systems approaches, Information, Non-linear dynamical systems, Non-linear dynamics, Open system, Pattern formation, Self-organization, State (phase) space, System