The type of complexity something must have to be a problem for evolution to explain.

Michael Behe is the one who coined this term, and he defines it as "a single system composed of several well-matched, interacting parts that contribute to the basic function, wherein the removal of any one of the parts causes the system to effectively cease functioning."

The definition proposed by Behe is not always workable. Study of the biology of various living things gives indications that many features of those living things have resulted from what might be called "jury-rigged design" - where something that was serving one purpose has mutated slightly and now either performs an additional purpose, or perfoms a new purpose instead.

True irreducible complexity would require finding a feature that not only stops working for the intended purpose* when one part is removed, but that serves no purpose whatsoever, and could not conceivably be adapted for a new one with minor changes.

* Technically, the term intended purpose makes no sense when discussion evolution, because there was no designer to create the feature for a specific purpose, it just happens to have evolved to perform whatever it does.

Unfortunately, this concept neglects the importance of reuse in evolution. One of Behe's examples is the bacterial flagellum that is a complicated machine that would not work without all of its parts. This line of reasoning ('it cannot work as anything less than 100% complete') is used to argue the impossibility of stepwise construction. Also known as the 'what use is half an eye' argument.

The simple solution to the question is that the parts were used for something else. That is to say, they once had a different function (perhaps an ion pump for the flagellum). This function was overridden by the organism to provide a new structure. This scavenging tendency is the reason for the homology found between genes in different branches of life. It is also why moonlighting proteins are relatively common (such as the crystallins).

Unfortunately the argument is based on the usual tyranny of probability. For example, consider a two component system : S = {C1, C2}. The principle of irreducible complexity says that the probability of the system evolving is:

P(S) = P(C1) * P(C2)
In words, this is (probability of the system evolving) equals (probability of component 1 evolving) AND (probability of component 2 evolving). If the components are improbable themselves, then the product of their probabilities is very small. Indeed, since proteins are often described using infinite monkeys, the product of several small numbers is very small indeed.

This is, of course, rubbish. It ignores the simple fact that systems evolve as a whole. Ironically, it is the very tendancy to assume things are instantaneously created that leads people to such assumptions. Nobody seriously thinks that systems like this jumped into existance without any predecessors. More importantly, the ancestor systems were fully functional.

A key point for Behe to think about : evolution can be irreversible. Just because a system seems unstable to change (irreducible, in other words) does not mean it couldn't be evolved. In other words, consider a sequence of systems (or {S1, S2...Sn} ) which increase in irreducible complexity over time. That is, they come to depend more and more on each other through interlocking changes in their components (C1 and C2).

Clear? :)

Recently, the concept of irreducible complexity coined by Michael Behe has become cause celebre for the creationist movement. There are several problems with Behe's proposal vis-a-vis creationism. First and foremost of which all of his examples are actually quite reducible. His model argument is the mousetrap which consists of 5 parts; spring, trigger, baitholder, jaw and base. None of which can be removed or the trap will not function according to Behe. With a little creativity a mousetrap can be made from the spring alone. Similarly Behe erroneously attibutes this property on other things.

Furthermore, many of the proponents of this argument from design are theologians and philosophers and few biologists reject evolution on this basis. There is also plenty of literature that undermines Behe's theory. Hume effectively refuted the argument from design over 200 years ago. Richard Dawkins renders Behe's assertion moot in a book called The Blind Watchmaker.

With respect to basic logic Behe uses a false dichotomy. Behe only considers two possibilities contigent upon the existence of irreducible complexity, namely evolution or creator.

Creationist who dismiss evolution also dismiss a preponderance of evidence such as analagous proteins and biochemical pathways that exist across phyla, not to mention established concepts like allopatry or fossil records.

Creationists have taken irreducible complexity up as a concept that further's their cause. What they fail to realize is that Behe himself does not deny evolution he allows for the assumption of the existence of an intelligent designer based on the existence of his theory. A theory that includes a property which has yet to be accurately attibuted to anything. This is not to say that a creator does not exist, merely that the concept of irreducible complexity is no basis for that assertion.