A common example of this was the bacterial flaggellum, which is a thick rod extruding from some bateria that is spun around to move them through their liquid habitat. The complexity of the flaggelum, which has about 80 moving parts that each require the other moving parts for the whole to function correctly could not have come about through darwinian selection, it was argued. For any one part to evolve needed the other parts in place - any one component could not evolve without the others, or the flaggellum would not work.
The flagellum example was popular until it was shown that other, more primitive forms of flaggellum exist - an intermediate, simpler form that gave a sort of snapshot of how evolution had created the finely functioning finished flagellum.
Blood clotting is the latest example. Blood clotting is a fantastically intricate process that requires the simultaneous interactions of hundreds of proteins. Therefore... yadda, yadda, you get the picture.
The problem with irreducible complexity is obvious. As the evolution of one complex piece of machinery or one convoluted biological process is partially explained, proponents can simply surf onto another. There are plenty of examples of complicated processes in biology, after all, and science is not so advanced as to be able to outline plausible evolutionary paths for all of them.
You see, the structures and processes we see now are, to us, the final product. Unlike larger features that can be traced in the fossil record, no stone-imprinted proof awaits our discovery of the evolution of blood clotting agents, there aren't always easily discoverable primitive forms of flagellum, because the prototypes have died off, their descendents inheriting the final, seemingly perfect form.
The sum of irreducible complexity comes down to "evolution is disproven because we cannot see it working", which is no satisfying argument at all.
Imagine a man climbing a cliff with only a short ladder. Fortunately, the cliff has many horizontal ledges set not too far apart that he can scale. He sets his ladder, clambers up, then lifts his ladder up after him. In this way, from ledge to ledge, he can scale the cliff, which is many times higher than his ladder. Someone, who hadn't observed the process, and who believes in irreducible complexity, might imagine that the man must have had outside help to scale the whole cliff. After all, his ladder is too short to reach the top. And there is no evidence of the ladder left behind on any of the ledges.