This is an idea primarily promoted by anti-evolution proponents in the United States and is not accepted by mainstream science. The basic argument of irreducible complexity is that some organs (or body parts) are so complex that without their subsystems, they would not have worked, and therefore someone (God) must have designed them as a whole without any evolution. Thus this is a subset of the Intelligent Design movement.

One instance of this may be found in a book by Michael Behe, Darwin’s Black Box (1996).  He discusses several biological systems that he maintains cannot be explained through evolutionary theories.  This claim is challenged by biologists who have successfully shown evolutionary roots of some of these systems.

For a description of irreducible complexity and the specific case of the Bacterium Flagellum, see this excerpt (9 mins) from a Nova episode:  https://www.youtube.com/watch?v=a_5FToP_mMY.  For the full version of the NOVA episode, “Judgment Day”, see https://www.youtube.com/watch?v=7HZzGXnYL5I.