Diwali, the festival of lights, is the main day of the five days of the festival. It falls on Amavasya, the new Moon day, of the month of Ashwina by the lunar months that end at the new Moon, which is Kartika by the months that end at the full Moon. On this evening families light lamps, decorate the home with rangoli, exchange sweets and gifts, and worship Lakshmi, often together with Ganesha. Different regions link the day to different stories, including the return of Rama to Ayodhya.
The day is fixed by the Pradosh, the period after sunset. Diwali is kept on the civil day on which Amavasya is in force during Pradosh, and the Lakshmi Puja is done in the evening, within Pradosh, in the muhurat when the rising sign is Taurus (Vrishabha). When Amavasya covers Pradosh on two evenings, the later evening is taken only if Amavasya covers at least about 24 minutes (one ghati) of Pradosh on it, and otherwise the earlier one. This page follows the Smarta conventions that most published calendars use, and the table is checked against a widely used panchang for five cities.
Why do cities differ? Amavasya begins and ends at one instant worldwide, but sunset is at a different clock time in each city. The new Moon can arrive before dusk on one date in a city in Asia and after dusk on the same date in a city in North America, so Diwali is kept on different days in different parts of the world in some years, and the puja muhurat differs by city every year. This is the reason for a city table rather than one national date.
A calm note: the heart of Diwali is light, family and giving, not the minute of a muhurat. If a window is hard to meet because of work or travel, a lamp lit with a sincere prayer in the evening is a complete observance. If your family or temple keeps a different day, it is right to follow them.