The idea of irreducible complexity as described by Michael Behe in his book Darwin's Black Box is that there are complex systems "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". This lack of a functioning precursor to a complex system means the individual parts must have arisen without the benefit foreseen in the final component. But natural selection requires each tiny step of change to be beneficial along the way.
We understand the benefit of the integrated complex systems which perform some amazing functions like vision or hearing or respiration. But how can the parts which make up this system be of any benefit by themselves? What good is a light sensitive spot (an early retina) without the nerve to carry the signal or without the brain function to interpret the signal as vision? Just as the individual parts of an airplane are all non-flying parts until they come together in an organized and specified complexity, the parts of a integrated system also perform no worthwhile function without the other parts.
The famous example Behe gives to illustrate this concept is the mousetrap. As simple as it is, the mousetrap is highly effective but irreducibly complex. The mousetrap is made up of five major parts as illustrated in the diagram.
How effective would this mousetrap be if it were missing any one of the parts? Would it be able to catch any mice without the holding bar for example? Of course the answer is that any missing part would render the mousetrap useless and essentially broken. Any precursor to an irreducibly complex system is by definition useless. All parts must come together at same time, and along the way, each separate part has to be beneficial to survival. If a part is not yet integral to the complete system, it must play some other beneficial function in the meantime (millions of years).
People have often talked about building a better mousetrap. Well, the "mousetraps" we have in our bodies such as the eye are models of perfection so the question is not how to make it better but how to make it at all.