Irreducible complexity, as defined by Michael Behe in Darwin's Black Box is a property of a system such that if any part is removed, the system ceases to function. Irreducible complexity is often used as an argument for Intelligent design.
The classic illustration of an irreducibly complex system is a mousetrap: it consists of a base, hammer, spring, catch (or trigger), and fasteners to hold the pieces together. If any of those parts is removed, the mousetrap no longer works.
The argument, then, is that since evolution proceeds by adding parts to an existing system one by one, the precursors of an irreducibly complex system would have been useless, and would not have been selected for. Ergo, all of the pieces had to be put together by an intelligent designer.
The argument that evolution always proceeds by adding parts is false. Natural selection can remove parts as well as add them. For instance, whales have no hind legs, but retain vestigial pelvises where their ancestors' legs were attached.
Another example of an irreducibly complex system is a gothic arch: if any stone is removed, the arch falls down. The way to build such a system is to install a scaffold, build the arch, then remove the scaffolding. Similarly, biological mechanisms do not have to co-exist with the structures that allowed them to evolve the way they did.
Incomplete systems can still have a function
While it is true that an irreducibly complex system with a missing part loses its original function, it may still have some other function. For instance, a mousetrap without a catch can still work as a tie clip, or a paperweight. A mousetrap without a base can be nailed to the floor. Such a mousetrap would not be as useful, but would still function.
For example, the bacterial flagellum, a long spinning hair that functions as an "outboard motor" for bacteria, is often cited as an example of an irreducibly complex system. But if some of its parts are removed, the resulting system bears a striking similarity to the Type Three Secretory System, a "syringe" that allows bacteria to infect other cells.