Short answer
Tamil New Year (Puthandu) is the first day of Chithirai, when the Sun enters sidereal Mesha (Aries), in April. Every Tamil month works the same way: it begins when the Sun moves into the next sign, so months run from mid-month to mid-month on the Western calendar. The Panchangam shows today's Tamil month; the Festivals tool gives this year's Puthandu for your city.
How ChiChakra reads this
ChiChakra finds the moment the Sun enters each sidereal sign and applies the usual Tamil rule: if the change happens before local sunset, the month starts that day; if after, it starts the next day. That is why Puthandu can fall on different days in Chennai and Toronto in the same year.
Reading the Tamil months
The twelve months follow the Sun's sign in order: Chithirai (Mesha), Vaikasi (Vrishabha), Aani (Mithuna), Aadi (Kataka), Aavani (Simha), Purattasi (Kanya), Aippasi (Tula), Karthigai (Vrischika), Margazhi (Dhanus), Thai (Makara), Maasi (Kumbha) and Panguni (Meena).
So the Sun's sign today tells you the month. In early November the Sun is in Tula, which makes it Aippasi; Karthigai begins once the Sun enters Vrischika later in the month.
Thai begins with Pongal, which is why Pongal and Puthandu both follow the solar rule rather than a lunar tithi.
Good to know
Kerala's solar calendar uses a slightly different rule for when a month begins, so a Malayalam month can start a day apart from the Tamil one.