How the date is fixed
Diwali falls on the new-moon day (Amavasya) that closes the lunar month Ashwina. The date is the day on which that tithi is running at pradosh, the twilight just after sunset. In the North Indian (purnimanta) calendar the same fortnight is counted in Kartika.
About
Diwali is kept on Amavasya, the new moon that ends Ashwina in amanta reckoning and Kartika in purnimanta, in October or November. The Lakshmi Puja is done in pradosh, the period after sunset, so the day whose evening carries Amavasya is chosen; when Amavasya is present at pradosh on two days, the second is taken, and when on neither, the day on which Amavasya covers most of the evening. Sthira lagna (Vrishabha rising) within pradosh is the window most almanacs print. Bengal and Assam keep Kali Puja at midnight of the same Amavasya, and the Jain tradition marks it as Mahavira's nirvana.
How it is observed
Lamps lit at dusk, Lakshmi and Ganesha worship in pradosh, new account books for traders, and sweets shared with neighbours.