Short answer
Your Hindu birthday is the day your birth Nakshatra (lunar mansion) or birth Tithi (lunar day) comes round again in your birth month. Tamil and Kerala families usually celebrate the Nakshatra in the solar birth month; many North Indian families use the Tithi in the lunar birth month. Both move against the Western calendar every year, so use the Hindu birthday finder.
How ChiChakra reads this
ChiChakra takes the Nakshatra and Tithi from your birth details, then finds the day in this year's matching month when that Nakshatra or Tithi is running at sunrise in the city where you will celebrate. Most families go by sunrise, though some customs differ when the star or tithi is very short.
Choosing your day
Decide which tradition your family follows before you look. The Nakshatra and Tithi birthdays can fall weeks apart, and neither has to sit near your passport birthday.
Use the city you will celebrate in. If you were born in Madurai but live in Manchester, your birth details stay as Madurai while the celebration city is Manchester, because local sunrise decides the day.
In a year with an extra lunar month (Adhika Masa), families differ on which month to use for a Tithi birthday, so ask your elders which they prefer.
Good to know
An accurate birth time matters: if you were born close to a changeover, a few hours can change your birth Nakshatra or Tithi.