Dark magic, or black magic, in most contexts is typically magic that is used or obtained through non-natural or malicious ways. Instances of this are things like necromancers, who commune with and control the energies of the dead; sorcerers who form pacts with demons to gain access to their magical properties; mages who actively destroy or consume the living or life forces of the world to produce their magic; and then there's a subset of dark magic in which an individual may use it due to a curse or exposure to the dark energy. This general concept still holds pretty true in this game.
A lot of Demacia considers magic in general to be dark magic. This is flawed logic on their part, because mages like Syndra or Lux are inherently born with intense magic power that is not derived from destructive forces nor is it by its nature malicious, such as Void magic or the life consuming magic employed by Vladimir. Also in the world are shamanistic characters such as Udyr or Ivern, who's magic is linked to nature, but is used to help and protect nature, and not drain it. These would not be dark magicians by normal standards, but most Demacians wouldn't care.
Vayne, by that line of thought, despises magic. As in a deep, triggering, hateful, loathing of magic. As in she is so entirely blinded by her vengeance and hatred that to her, there is nothing but black magic in the world.
