There are many popular stories associated with this day, but the most widely accepted one is that of Satyabhama (Krishna's wife) and Narakasura. Narakasur was a demon king, ruling Pragjothishyapur, a southern province in the present day Nepal. He gained a boon from Brahma that he shall die only in the hands of a woman. Armed with the boon, he became a cruel king. Narakasura was infamous for his wicked ruling and high disregard for gods and women.
Addicted to power, he defeated Lord Indra (king of gods) and stole the earrings of Aditi (the heavenly mother goddess). Aditi was a relative of Satyabhama. When she heard of the injustice being done to women in general by Narakasura and his behaviour with Aditi, she was enraged. Satyabhama went to ask Lord Krishna, permission to wage a war against Narakasura, Krishna not only agreed, but also offered to drive her chariot in the Warfield.
On the day of the war, both the armies fought bravely and the war continued for some time. Satyabhama fought Narakasura bravely, but she was no match to his trained war wisdom. After some days, when Narakasura got a chance, he took an aim at Krishna, hurting him lightly. Krishna fainted and made Satyabhama furious. She doubled her attack on the demon king and killed him finally. Her victory on Narakasura translated into freedom for all his prisoners and honoring of Aditi.
To announce the death of Narakasura, Krishna smeared the demon's blood on his forehead and returned the very next day along with Satyabhama to his kingdom. On their arrival, preparations were made to cleanse Krishna of the demon's blood. At dusk, the whole city was lit with lamps and fireworks were displayed to rejoice in peace after the death of the demon king. Thus, came Narakachathurthasi as a celebration of the death of the evil king.