Every master has an apprentice for otherwise they are a master of nothing, and the Dark Apostle is no different. Even with all the bio-augmentations, power armour, daemonic favours and an assortment of cheat codes, a Dark Apostle can't be everywhere at once. Hence, this is where the Dark Disciple comes in. Knowing the backstabby nature of Chaos (yes we know Khorne would mostly stab you in the face) it is a pretty smart incentive to have your followers and students consist entirely of meek and feeble normies than it is fellow superhuman supersoldiers, barring one or two actual heirs.
A Dark Disciple is a special type of human Chaos Cultist that follow their Dark Apostle masters into battle. They often carry dark relics to help him in his dark arts or practice his sermons in the midst of battle. During peacetime, they assist him in creating sacrifices and/or any other religious activities. Basically, your daemonic choir boys to your daemonic catholic priests. Although how touchy these priests are is anyone's guess... unless they're more Slaaneshi-orientated, then it is a prerequisite to be touchy.
On tabletop, they function as the Chaos version of the Watchers in the Dark, which is the non-combative utility-based unit that significantly helps the Dark Apostle in many ways, one of which is to simply take a pair of Dark Disciples with your Dark Apostle for a handy boost to your rolls.