The eye
Irreversible complexity is a term referring to the fact that certain structures (such as the human eye) are so complex, that it is incomprehensible to imagine how they could have evolved by gradual changes over time: the complexity cannot be reversed back to simplicity. This is true of other complex biological structures and is one of the arguments of the Intelligent Design movement. This organisation isn't Christian based, it simply states that design in the Universe suggests a Creator.
At the back of the human eye is the retina, which is less than one square inch yet contains over 137,000,000 light sensitive cells all connected to your brain.
"It would take 100 years of Cray [super computer] time to simulate what takes place in your eye many times every second" Reverse Engineering the Brain, 1985.
The blood cells are in front of the retina in the human eye to act as a last line of defence against UV light. In an octopus they are behind the retina, which makes the eye more effective as the water filters out the UV, so their eyes are protected by their environment, in contrast to humans. How did one type of eye evolve into the other?
Another example of Irreversible complexity and the eye is the owl, which has a cylinder shaped eyeball that cannot rotate in its socket. This is why it has to move its whole head to look in different directions. How did this evolve? And how did it see when it was part way between a sphere and cylindrical shape?
Psalm 94:9 "Does he who implanted the ear not hear? Does he who formed the eye not see?"
Further reading: The Miracle of the Eye
Darwin himself stated "If it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down. But I can find out no such case." (p187-190 Origin of the Species, 1859). At the time, there was only a basic understanding of the eye and other complex structures such as the brain.
Darwin expressed his own misgivings on his evolutionary theories in his book, "The Origin of the Species" as he contemplated irreversible complexity and the eye. "To suppose that the eye could have been formed by natural selection seems, I confess, absurd in the highest degree." Today we realise that so many organisms demand the highest level of complexity for the minimum requirement for their functions. Later in his life, Darwin called much of the content of his evolutionary theory "the unformed ideas of my youth."
Another example of complexity in Biology is that some bacteria have a complex motor that rotates up to 100,000 rpm and drives a cord (flagellum) which acts as a "propeller" so that the bacteria can swim! The motor is so tiny that 8 million of them would fit on the cross section of a human hair. The amazing motorised germ
Complex living things cannot evolve piece by piece. Have a look at the book Darwin's Black Box, by Dr Behe.