She achieved this feat in the Battle of the Pelennor Fields, during which the Witch-king, seemingly invincible, had just slain King Théoden. Eowyn, disguised as a man and riding alongside her cousin Merry, challenged the Witch-king.
Despite the Witch-king's power and fearsome reputation, Eowyn managed to pierce his armor with her sword, aided by Merry's unexpected stab with a barrow-blade. This mortal wound, combined with the breaking of the Witch-king's power through the destruction of the One Ring, ultimately led to his demise.