Short answer
Nakshatras are the 27 lunar mansions of Vedic astrology, each covering 13°20′ of the zodiac. Your birth star (Janma Nakshatra) is the one the Moon occupied when you were born. It's used for naming syllables, marriage matching, your Hindu birthday and the start of your Dasha sequence, and tradition gives each star its own symbol, deity and temperament.
How ChiChakra reads this
We take the Moon's sidereal position (Lahiri) at your birth and find the Nakshatra and pada (quarter) it falls in. The Moon spends roughly a day in each Nakshatra, so two people born a day apart usually have different birth stars even when they share a Rashi.
What your birth star is used for
Dasha timing: the ruling planet of your Nakshatra sets which planetary period you were born into.
Matching: Porutham and Ashta Koota both compare the partners' birth stars.
Hindu birthday: many families celebrate when the Moon returns to the birth Nakshatra in your birth month of the Hindu calendar.
Daily life: panchangams list each day's Nakshatra, and many people check how it sits with their own. Read the traditional temperament as a reflection prompt, a lens rather than a verdict.
Good to know
Near a Nakshatra boundary, a small birth-time error can change your star, so use a recorded time where you can.