What it means
The 27 nakshatras divide the same 360 degrees the twelve signs do, into arcs of 13°20' each. Where a sign is a broad temperament, a nakshatra is a much finer address, and Vedic astrology leans on it for things the signs are too coarse to settle. Your birth star is the nakshatra the Moon occupied at birth.
How it is read
It does more work than most newcomers expect. The Vimshottari dasha sequence starts from the Moon's nakshatra, so it sets the entire timeline of a life. Guna Milan compares two Moon nakshatras and nothing else. KP subdivides the nakshatra again to reach its sub-lord. Each is a different technique resting on the same 27 divisions.
What to watch for
Each nakshatra splits into four padas of 3°20', and the pada fixes the syllable a child is traditionally named from. Because the Moon crosses a nakshatra in about a day, the birth star is one of the few chart facts a rough birth time will usually still get right — the pada, at a quarter the width, is not.