The Legendary-Tale associated with the Navratri tells the story of the great battle fought between a very powerful Demon namely, Mahishasur, and the Goddess Durga. Actually, Mahishasur by worshipping Lord Brahma had attained a unique blessing of immortality from him, with one condition that he could be defeated only by a woman. Having this blessing of immortality & powers, the demon attacked the Triloks- the earth, the heaven & the hell and defeated everyone whosoever came across him. Because of the blessing, even the Gods could stand a chance against him on the battlefield. The Gods then prayed to Lord Brahma, Lord Vishnu & Lord Shiva to help them in defeating their powerful enemy.
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Keeping in view the helplessness of the Gods, Lord Vishnu ultimately decided to create a woman to overpower the demon, Mahishasur. After the creation of the woman who was named 'Durga', the Gods approached the God of Destruction, Lord Shiva to seek his help. Being the most powerful God, Lord Shiva not only blessed them with fearlessness, but He along with Lord Brahma put all their powers together in Goddess Durga to empower her to kill the demon.
Since it is also believed that Goddess Durga is actually a Reincarnation of the Goddess Parvati, the wife of Lord Shiva, as such Durga or the Shakti- another Avatar of Goddess Parvati-is the Goddess of Power that runs throughout the universe forever.
After the creation of Durga by the three most powerful Lords-Brahma, Vishnu & Mahesh(Shiva), she fought a 15-Days Long battle with Mahishasur. This fight was so fierce that it shook the Triloks- the earth, the heaven & the hell. During the fight, the clever demon kept on changing his forms and disguised into different faces just to confuse and mislead his opponent Goddess Durga. Finally, when he took the form of a Buffalo, Goddess Durga pierced his chest with her Trishul(A fork-shaped weapon) and killed him instantly. Thus, it also signifies the victory of the Good over the Evil.
In Hindi Mythology, there are actually Four Navratri in a year, out of which, Two Main Navratri are celebrated in the months of 'Chaitra' & 'Ashwin' of the Hindu Calendar whereas the other two Navratris which are also termed as the "Gupt Navratri"(Also especially known for performing 'Tantra Pooja') are held during the months of 'Magh' & 'Aasharh' every year.
According to the rituals during all nine days of Navratri, a different Avatar of the Goddess Durga is worshipped and people observe fast for all the nine days and on the last day, break their fast after symbolically worshipping the 'Kanyas'- little unmarried girls. Also, all the Navratris incidentally come at the times of Season's Changing, thereby signifying the changeover of the climate every year.