Special Children refers to children to whom Azazel fed his blood when they were babies, having snuck into their homes after making a deal with their parents.
The result was that the blood awoke/created psychic abilities in them so they could (apparently) lead an army of demons in a war against humanity and God.
Around Season 2, Azazel wanted to test these children to find who would be the strongest, so he put them in an abandoned town called Cold Oak and told them only one could survive. This eventually triggered a fight among the special children. Azazel led everyone to believe that he wanted a leader for his demon army.
However, as it was later revealed, the endgame was bigger than just leading a demon army. Azazel chose the children because they were all potential vessels for Lucifer. The strongest one would be chosen to contain Lucifer throughout the apocalypse. Sam Winchester was the sole survivor and so, despite his greatest efforts, was taken over by Lucifer.