The worship of Devi Durga in the month of Ashwin is called 'Akalbodhan'- an uncoventional time for inauguration of the worship. It is so called since the period of this worship differs from the conventional period (during the spring - 'Basanta').
In the ancient times, a demon called 'Mahishasura' earned the favor of 'Lord Brahma' through extreme austerity and prolonged meditation. Pleased with the devotion of the demon, the lord blessed him with a boon that no man or Deity would be able to kill him. Empowered with the boon, Mahishasura started his reign of terror over the Earth and Heaven.
Invasion of Heaven:
People were already being killed mercilessly and even the Gods were assaulted.
Yet the worse was yet to come. It came when an strong army of demons was gathered to siege the abode of the Deities. The army was led by the king Mahishasura, the green skinned demon with the form of a giant buffalo. With its weapons of iron, and its phalanxes of elephants and charioteers the army finally marched on the king of gods, Purandara or (Indra), defeating him. Then, Mahishasura usurped the throne of heaven.
The victory was complete, and all the gods were driven out of the heaven. Routed they went to the trinity of the Supreme Gods, Brahma, Vishnu, and Shiva to save themselves and the men on the world.
The orgy of violence vis-a-vis the the invasion of the heaven by the evils enraged the Supreme Gods. Their faces were flushed with anger and a dazzling luminescence flooded forth. Great flames and thunderbolt streaked through all directions. The fires illuminated all the three worlds: the heaven, the earth, and the nether-world with penetrating beams of light.
The Birth of Durga:
At a single point, the energy of all the fires coalesced and assumed a shape, in the form of a young woman. Her face was from the light of Shiva. Her ten arms were from Lord Vishnu. Her legs were from Lord Brahma.
The dispossessed gods were awed by the supreme energy and enchanting beauty of the conceived Goddess. They praised her and equipped her with their divine gifts: Shiva gave Her a trident with a spear-end. Krishna gave her rotating disc. Varuna, the God of sea, gave her a conch and the God of fire gave her a missile. From the wind, Vayu, she received arrows. The king of gods, Indra, gave her the thunder-bolt, and the gift of his white-skinned elephant Airavata was a bell. From Yama, the god of death, Durga received a rod, and from the Ruler of Waters she was given a noose.
Durga received many other precious and magical treasures-- gifts of jewels, new clothing, and a garland of immortal lotuses for her head and breasts. Heaven's architect gave her a bright axe and magic armor. God of mountains, Himalayas gave her jewels and a magnificent lion to ride into battle.
Now equipped with the fearsome weaponry and magical powers of the gods, and dressed in golden armor and jewels she set off, seated gracefully upon the lion. His thunderous roars shook the three worlds. Oceans swelled up to scrape the sky and surf broke over the land. Continents were torn at their granite foundations as whole new chains of mountains rose, while older ranges crumbled, cracked, and gave way to dust in a thousand landslides. Seeing these cataclysmic ripplings in waves through all the three worlds, Mahishasura and his demon allies found their attention drawn from heaven to Earth. Though confident of their power and control in heaven, even the conquering demon host could not help being awestruck.
The demons had little time to admire the radiant visage of their new adversary, for soon she engaged them on the battlefield. First, the army of Chikasura, and then that of Chamara, the two chief commanders (also called Shumbha and Nishumbha,of Mahishasura by some) were met. They were destroyed in a great battle. Now it's turn for Mahishasura.
The Termination of Mahishasura:
Confident but confused by the humiliating defeat of his loyal and powerful commanders Mahishasura did his best in arranging and equipping his personal army.
This time the heaven was led by Durga as the boon of Mahishasura could only make him invincible against all but woman. Surrounded by chants of praise, the blowing of horns the beating of drums and songs of worship Durga roamed the battlefield on her mighty lion. From her divine breath her army was constantly replenished with new warriors, each able, brave and resolute.
Shocked and enraged by the disastrous events on the battlefield. In a mad desperate bid Mahishasura then reverted to his own form, a buffalo, and charged about on the battlefield. In a wild rage he charged at Durga's divine soldiers wounding many, biting others and all the while thrashing with his long, whip-like tail. Durga's lion, angered by the presence of the demon-buffalo, attacked him. While he was thus engaged, Durga threw her noose around his neck.
But through magical spell Mahishasura kept changing his shape and form from one to another so as to puzzle the Devi.
Finally the Goddess beheaded the buffalo and from it emerged Mahishasura in his original form. Durga pierced his chest with the trident and relieved the world from the evil power.
The gods returned to heaven, and along with the sages of the earth, they sang praises and showered floral compliments to the Goddess Durga. Henceforth, and to this day, the Goddess Durga is worshipped by all the gods in heaven, and all human beings on earth. Mahishasura is there too--frozen in his moment of final defeat, impaled by Durga's spear and seeking pardon beneath her left foot. The Goddess Durga then continued to be worshipped in this form.
According to Puranas (the epics) , King Suratha, used to worship the goddess Durga in spring. Thus Durga Puja was also known as Basanti Puja (Basanta being Spring). While the vernal worship of Durga still goes on but it is the Ram's Akalbodhan during the autumn that came to be a most widely accepted practice.
In the 'Ramayana', as it goes, Rama went to 'Lanka' to rescue his abducted wife, Sita, from the grip of Ravana, the king of the Demons in Lanka. Before starting for his battle with Ravana, Rama wanted the blessings of Devi Durga . He came to know that the Goddess would be pleased only if she is worshipped with one hundred 'NeelKamal' or blue lotuses. Rama, after travelling the whole world, could gather only ninety nine of them. He finally decided to offer one of his eyes, which resembled blue lotuses. Durga, being pleased with the devotion of Rama, appeared before him and blessed him. The epical battle started on the 'Saptami' and Ravana was finally killed on the 'Sandhikshan' i.e. the crossover period between Ashtami (the next day) and Navami (the day after). Ravana was cremated on Dashami. This is why Dashera is celebrated in India with so much of fanfare and the effigy of Raavana is burnt.
In course of time Bengalis adopted the autumnal worship of Durga performed by Rama and made it their main festival. The Pujas span over the four days, the time taken by Rama to finally kill and cremate Ravana.