Irreducible complexity describes any system which requires all of its parts in order to perform any function, even not its main or intended one, and where the removal or alteration of any part would prevent the system from performing any function whatsoever.
In the origins debate, the assertion that various organs of cells or animals are irreducibly complex is used to support the theory of Intelligent Design or Creationism. Proponents of Inteligent Design or Creationism argue that irreducible complexity is evidence of an intelligent agent.
William Dembski states the following in his book No Free Lunch:
"A system performing a given basic function is irreducibly complex if it includes a set of well-matched, mutually interacting, non-arbitrarily individuated parts such that each part in the set is indispensable to maintaining the system's basic, and therefore original, function. The set of these indispensable parts is known as the irreducible core of the system. 
In Intelligent Design
Irreducible complexity has been used by astronomer Fred Hoyle and biochemistry professor Michael Behe to argue against the idea of evolution through natural selection. Hoyle compares the probability of all the parts of a cell coming together through natural forces to the probability of a cyclone in a hangar assembling all the parts of a Boeing 747 aircraft.
William Dembski wrote the following in an essay:
|“|| In a 1996 review of Michael Behe’s book Darwin's Black Box, James Shapiro, a molecular biologist at the University of Chicago, wrote: "There are no detailed Darwinian accounts for the evolution of any fundamental biochemical or cellular system, only a variety of wishful speculations. It is remarkable that Darwinism is accepted as a satisfactory explanation for such a vast subject -- evolution -- with so little rigorous examination of how well its basic theses work in illuminating specific instances of biological adaptation or diversity" (National Review, 16 September 1996).
Five years later cell biologist Franklin Harold wrote a book for Oxford University Press titled "The Way of the Cell." In virtually identical language, he writes: "There are presently no detailed Darwinian accounts of the evolution of any biochemical or cellular system, only a variety of wishful speculations." 
In addition, Michael Behe wrote the following:
|“||Molecular evolution is not based on scientific authority. There is no publication in the scientific literature—in prestigious journals, specialty journals, or book—that describes how molecular evolution of any real, complex, biochemical system either did occur or even might have occurred. There are assertions that such evolution occurred, but absolutely none are supported by pertinent experiments or calculations. Since no one knows molecular evolution by direct experience, and since there is no authority on which to base claims of knowledge, it can truly be said that—like the contention that the Eagles will win the Super Bowl this year—the assertion of Darwinian molecular evolution is merely bluster." - Michael J. Behe, Darwin’s Black Box (New York: The Free Press, 1996), p. 186||”|
Intelligent design theorists agree with Charles Darwin that a biological system could not make a gradual transition via small successive steps, unless each step provided an advantage to the species.
Behe presents several examples of organs and cells which he says could not have evolved through natural forces alone due to their irreducible complexity.
Behe introduced and defined this term on page 9 of his work, Darwin's Black Box:
- Irreducible complexity is a "single system which is composed of several interacting parts that contribute to the basic function, and where the removal of any one of the parts causes the system to effectively cease functioning."
It is therefore a state of interacting components that could not function without all its parts. In particular, the term is used by proponents of intelligent design to argue that irreducibly complex biological systems cannot evolve. However Behe himself had to admit that the argument of irreducible complexity does not fully disprove evolution:
|“||However, commentary by Robert Pennock and others has made me realize that there is a weakness in that view of irreducible complexity. The current definition puts the focus on removing a part from an already-functioning system. Thus, seeking a counterexample to irreducible complexity, in Tower of Babel Pennock writes about a part in a sophisticated chronometer, whose origin is simply assumed, which breaks to give a system that he posits can nonetheless work in a simpler watch in a less demanding environment. The difficult task facing Darwinian evolution, however, would not be to remove parts from sophisticated pre-existing systems; it would be to bring together components to make a new system in the first place. Thus there is an asymmetry between my current definition of irreducible complexity and the task facing natural selection. I hope to repair this defect in future work.||”|
The most commonly used examples of irreducibly complex biological systems are the human compound eye, the blood clotting cascade, and the bacterial flagellum. Kenneth R. Miller describes hypothetical precursors to the flagellum's being used as ionic channels within bacteria, known as the Type III Secretory System. However, Professor Behe counters that Miller's criticism of his work is flawed.  Critics, including biology professor John McDonald, have countered Behe's toy model, the mouse trap, used to illustrate the concept. McDonald produced examples of how he considered the mousetrap to be "easy to reduce", eventually to a single part. However, Professor Behe disputes the criticism of John McDonald. 
- William Dembski, No Free Lunch,(2001), p. 285. ISBN 0742512975
- Reply to My Critics: A Response to Reviews of Darwin’s Black Box Michael Behe, 2001
- The Flagellum Unspun