Dhanteras is the first day of the five days of Diwali. It falls on the thirteenth day (Trayodashi) of the dark half of the month, in Ashwina by the lunar months that end at the new Moon and in Kartika by the months that end at the full Moon. The name joins dhan, wealth, with teras, the thirteenth. Families clean and decorate the home, light lamps at the door, and worship Lakshmi, Kubera and Dhanvantari, the physician of the gods.
The date is fixed by the Pradosh, the period after sunset. Dhanteras is kept on the civil day on which Trayodashi is in force during Pradosh (if it covers two evenings, the later one is taken only when it covers about 24 minutes or more of Pradosh), and the puja is done in the evening, within Pradosh, in a muhurat when the rising sign is Taurus (Vrishabha), a sign traditionally associated with steady prosperity. The window in the table is that muhurat for each city.
Why do cities differ? Trayodashi starts and ends at one instant worldwide, but sunset falls at a different clock time in each city. A tithi that ends shortly after dusk in one city may have ended before dusk in another, so the festival and therefore the following days of Diwali can differ by a day. The rising sign changes about every two hours and depends on the latitude and longitude, so the muhurat differs from city to city.
A calm note: Dhanteras is about welcoming care and plenty into the home, and there is no need to buy anything to keep it. If you choose to make a purchase, let it be something you need and can afford. If your family or temple keeps a different day or hour, follow them.