The story
The nine nights are the Goddess's. Mahiṣāsura, the buffalo demon, had a boon that no man or god could kill him; the gods pooled their powers into a woman, gave her their weapons, and she fought him for nine nights and killed him on the tenth. The Devī Māhātmya, seven hundred verses read through the festival, is the account.
Three sets of three: the first three nights are for Durgā, who destroys, the middle three for Lakṣmī, who provides, and the last three for Sarasvatī, who teaches. Some keep nine forms of Durgā, one a night (Navadurgā).
In Bengal the festival is Durgā Pūjā, and the story is a daughter's visit home: Umā comes to her parents' house with her four children for five days, and on the tenth the city weeps to send her back to Śiva on Kailāsa. Kolkata's paṇḍāls, some of them the size of temples, were inscribed by UNESCO in 2021.
