Epic fights are a new bonus in sigil-based events (with the exception of faction events), i.e. Invasion, Raid Boss, Tower of Doom, Class events and Bounty events.
You can give your troops a temporary boost by spending extra sigils and thus boosting their stats for one battle.
Before each battle you get the option to change the battle from Normal to Epic or higher (in the middle of the screen, right above Troop Restrictions). Changing to Epic costs an extra sigil, to Legendary 2 extra sigils, to Mythic 4 extra sigils.
Edit: One change is free (the first time you ever use this bonus, it doesn't cost you an extra sigil)
Just win a battle set to Epic or Legendary or Mythic and the achievement will unlock.