The Conception is an apocalyptic event which throws the world it affects into a temporary state of ruin, where it may be reborn in many different ways.
As is implied by its name, the Conception's purpose it to bring forth a new world, with new laws determined by a Reason. The Conception is set in motion by Hikawa, who strives to create a world of perfect stillness. After being set in motion, everyone outside of the hospital is killed instantly (except Dante) and turned into Magatsuhi or souls. Tokyo is uprooted and twisted into a spherical world known as Vortex World. The results of the Conception are determined by the actions of the Demi-fiend.
It's stated in partial declarations by Yuko Takao, Hikawa, and the Lady in Black that every universe has a finite amount of power, and once that power has run out, the only way to replenish it is for that universe to undergo the Conception in order to be reborn. Worlds that fail to initiate the Conception fall to the Shadow Vortex, and the Great Will, through Kagutsuchi, seeks to manipulate the process to its own ends. Lucifer apparently also has a hand in the manipulation through his selection of the Demi-Fiend.