The irreducible complexity argument was developed by Michael Behe and first used in his book Darwin's Black Box.
The argument is that some biological systems are so complex that they can not have evolved though mutation and natural selection; it you remove one peace, it brakes, it no longer functions. Creationists often use the analogy of a mouse trap; remove one part and it doesn't work. A mouse trap is surprisingly a good analogy, take out the catch and the hold down bar and you you have strong paper clip or if your defending evolution is Dover, a tie clip. Hell if you remove the spring and the hammer you have a paper weight.
One of the most popular examples of irreducible complexity is the eye. It is so popular because of a quote from The Origin of Species that would seem to say that the eye could not have evolved.
To suppose that the eye, with all its inimitable contrivances for adjusting the focus to different distances, for admitting different amounts of light, and for the correction of spherical and chromatic aberration, could have been formed by natural selection, seems, I freely confess, absurd in the highest possible degree
However this is a quote mine. Darwin continued with:
Yet reason tells me, that if numerous gradations from a perfect and complex eye to one very imperfect and simple, each grade being useful to its possessor, can be shown to exist; if further, the eye does vary ever so slightly, and the variations be inherited, which is certainly the case; and if any variation or modification in the organ be ever useful to an animal under changing conditions of life, then the difficulty of believing that a perfect and complex eye could be formed by natural selection, though insuperable by our imagination, can hardly be considered real. How a nerve comes to be sensitive to light, hardly concerns us more than how life itself first originated; but I may remark that several facts make me suspect that any sensitive nerve may be rendered sensitive to light, and likewise to those coarser vibrations of the air which produce sound
Then the fact that we have a pretty good understanding of the evolution of the eye makes the claim even more disingenuous.
Another popular example of irreducible complexity is the bacterial flagella. On the surface its would seem like a good example, it certainly looks designed however it is not. Remove a few parts and you get a type III secretory system: a type III secretory system is sort of like a tiny molecular syringe that injects toxins into other cells.
And last example in this post: wings. Its seems like a legitimate point; "whats the use of half a wing?" But with a little bit of imagination and some research you'll find that half a wing has benefits.
The most obvious benefit is gliding. Being able to glide would increase the distance an animal can jump and it would be especially beneficial to animals who live in trees. Being able to glide would make it easier to escape from predators and easier to capture pray.
Before any one decides to try and stump me with some example of irreducible complexity, use a little imagination and do some research. If you don't I'll use some imagination and some research to show you that you are wrong.
Links & Sources
Irreducible Complexity
Eye
Bacterial Flagella
Wings
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