Narakasura ruled from Pragjotishyapur (the present Assam region). Everyone including the Devas of Indraloka (the heavenly abode) suffered under his tyranny. He used to abduct women and behave harshly with them. The frustrated beings prayed Lord Krishna to redeem them from the tyranny of Narakasura.
Lord Krishna who is the 8th incarnation of Lord Vishnu, is omniscient.
When the atrocities of Narakasura crossed the limits, Lord Krishna goes to fight with Narakasura in His chariot with Satyabhama, His 3rd wife, as the charioteer. In the fierce fight between Krishna falls unconscious when Narakasura hits Him with the power packed arrow. On seeing her husband in an unconscious state, Satyabhama becomes furious. She takes the bow and arrows and fights bravely with the powerful asura. Finally, Satyabhama slays him in the fierce battle.
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Thus, Narakasura was killed on the day before Deepavali Amavasya i.e on bahula chaturdasi (14th day after the full moon day) in the Hindu month of Aswayuja, in the Dwapara Yuga. People rejoiced his death and celebrated the day by lighting oil lamps and with fireworks.
The following day i.e Amavasya day, people celebrated Deepavali with joy and fervor.
There are other versions of the story which are not so popular.
Other versions of the storyIn a different version which is popular in North-East India, King Narakasura wanted to marry Kamakhya Devi (in Assam). Kamakhya Devi says that She would marry him if he could lay down steps from the foot of Nilachal Hill (in the Western part of present Guwahati) to the temple. (Kamakhya temple is one of the Sakthi Peethas). The goddess puts the condition that the task should be completed within one night before it is dawn i.e the cock crows.
Narakasura takes up the mammoth task was almost done half way. When the goddess knows about this, She creates an illusion of dawn, before it was dawn and the cock crows. The asura thus leaves the task incomplete and was killed by Goddess Kali on the Chaturdasi day.
Naraka Chaturdasi is known as Kali Chaudas in the North / North-Eastern parts of India.
The other lesser known storyLord Vishnu takes the form of a wild boar i.e Varaha Avatara, kills the demon king Hiranyaksha, gets Mother Earth on His tusks and re-orbits Her. It is believed, Narakasura was born to Bhudevi and Varaha Avatara Vishnu. Hence, Narakasura is also known as Bhaumasura.
When the atrocities and wicked deeds of the demon crossed the limits, Lord Vishnu in Krishna Avatara, beheads the demon with His Celestial Sudarshana disc. In the battle with Narakasura, Krishna kills Mura who was the army general of Narakasura. Thus, Lord Krishna got the name 'Murari' (Sanskrit word 'ari' means enemy) i.e the one who is the enemy of Mura.
People, thus, rejoiced the death of the wicked and celebrated the day by lighting oil lamps and fireworks as a symbolic of driving away darkness from our minds and enlightening them in the spiritual path.
Thus it has become a customary to celebrate Naraka Chaturdasi / Kali Chaudas, a day before Deepavali Amavasya.