How nakshatra is calculated
The Moon’s position in the sidereal zodiac determines its nakshatra: one of 27 equal divisions, each spanning 13 degrees 20 minutes.
The nakshatra on a daily page describes the part of the zodiac occupied by the Moon at local sunrise. The accompanying time shows when the Moon moves into the next nakshatra.
Nakshatra and pada
Each nakshatra has four padas, or quarters, of 3 degrees 20 minutes each. A birth nakshatra and pada are calculated for the time of birth. They are different from the daily reading shown for today's sunrise.
Why ayanamsha matters
Ayanamsha is the offset between the tropical and sidereal zodiacs. It is applied to the Moon's longitude before identifying the nakshatra. Different ayanamsha conventions can therefore give different boundary times.
The method used here
This site uses the Lahiri ayanamsha, the one most Indian panchangams use.