Aura Break inverts the effects of Fairy Aura and Dark Aura: instead of multiplying the power of Fairy- or Dark-type moves by those Abilities' normal multiplier of approximately 4/3 (or more precisely, 5448/4096), the power of moves of that type is divided by 4/3 instead, resulting in a 25% decrease in power as long as both Abilities are active.
Even if a Pokémon with Mold Breaker, Teravolt, or Turboblaze uses a Fairy- or Dark-type move, the effect of Aura Break still applies.
Outside of battle
In Generation VII, Zygarde with the Ability Aura Break can be separated back into Cores and Cells by the Reassembly Unit.
If there is a Pokémon with the Dark Aura or Fairy Aura Ability, the Ability's effect will be reversed! Dark- and Fairy-type moves become weak!
Trivia
In Pokémon Super Mystery Dungeon, this Ability is practically useless, as Zygarde is not fought as an enemy, nor can Xerneas or Yveltal be fought using Zygarde, and it is impossible to even use Zygarde on the same team as Xerneas or Yveltal unless one is summoned with Djinn's Bottle.