Hindu Calendar demystified

The Vedic Chandramaana Lunar calendar system uses two Moon based properties.

First, to keep a day count clock in which Moon is the pointer(Clock Arm) and the stars are numerals in the sky Watch-Dial, pointed to by moon each day.
The Vedic astronomers, used the 13 degree movement of the moon between successive days and named the 27/28 stars pointed to by moon on a daily basis as 27/28 Nakshatra.
Simply a Nakshatra corresponds to moon position relative to Stars in one (solar) day.

Second is the Waxing-Waning of the  moon, can indicate a day-count or a moon day or thiti. There are 30 thiti(s) and are defined in a lunar month,  (each thiti being smaller than a solar day.) Fifteen are identified as Shukla paksha or ascending fortnight and next fifteen are called Krishna paksha or descending fortnight.

Source http://www.VedicAstronomy.net
Dr. Balakrishan.

This system of a lunar pointer as the primary calendar dates to the Vedic periods.

Special Sun-Days

The solar events Aayanas(Solstice s), and Vishuwat-Sankramana's (Equinox es)

Days are Long, then Days Become Equal to night then Days become shorter than night, then becone equal and then longer agsin, thats what we calls seasons
Every year earth travels around the Sun, in a fixed Elliptical(A circle with 2 Centers) path. So at 4 times the points of change in the periodic Solstices when apparent North-South movement of Sun reverses, usually occurring on June 21 and Dec 22.

Vishuwat means equal or the spring and fall equinox’s when daytime is equal to nighttime, usually occurring on March 21 and Sep 21. There are vedic references to solar singularities with corresponding solar/lunar pointed star locations.
Pournamsya, a time at which earth, sun, moon are aligned is a time of singularity used for religious purposes and formed the unit of half a month and is used in Rigveda.

The six-season definition is unique to vedic system and is not found in any other recorded culture or system. These seasons are, Vasanta, Geeshma, Varsha, Sharad, Himavanta and Shishira each season being about two moon cycles. It is in Taittareeya Samhita (Krishna Yajurveda) and in Atharva samhita 19th kaanda /7th Sootra that an explicit first definition and identification of the twenty- seven(28) Nakshatra’s is available (Refs, 1 and 2). It should be noted that the concept of Zodiac/Raashi is not even hinted in any vedic texts of the old period.

The table below provides a list of the twenty-seven stars from Taittareeya Samhita and Krishna Yajurveda 4th Kaanda 4th Prashna of Andhra School. Similar list is referred to in Atharvaveda, 19th kaanda/7th Sookta. It differs from the Yajurveda list in that twenty eight stars are listed. In 19th Kaanda/8th sooktha twenty eight(ashtha vimshaani) nakshatras are declared. The nakshatra not explicitly used in Jyotishya and in yajurveda is named Abhijit. The taiaareeya brahmana (third Ashtaka) derived from yajurveda again lists 28 nakshatras including Abhijit. It is referred to in Athrvana veda. The author or Drashtaara of Atharva veda sookta is Gaargya Rishi.

The confusion about 27/28 Nakshatras can be analyzed as follows. The 27.3 days taken by moon for visiting the same star can be rounded to either 27 or 28. Each of these integers represents a nakshatra or a daily star. It is possible that initially 28 nakshatras were proposed and defined to represent every day. This is evident from Gargya’s Nakshtara sooktha. Subsequently many years later when Jyotishya was taking a more formal and mathematical shape, 27 integer identities were perhaps found to be more reasonable and accurate. Hence one of the original 28 nakshatras had to be deleted. This could not be done easily as the vedas are considered to be ‘apourusheya’ and no liberties were allowed. Hence perhaps Abhijit, was declared to be an imaginary nakshatra meant only for phala.(ref 10).

The Table below lists names of Nakshatras, the deity to which Nakshatra is dedicated to, and alternate names. Ashwini and Bharani are listed as last two Nakshatras. Note that the list starts from Krittika and not from Ashwini as is currently used in Jyotishya Shaastra (Ref 3). The time at which Rishi Gargya documented the list of nakshatra's in Atharvana Veda is at least circa 2400 BC, as analyzed in appendix I. The appendix also dates the Jyotishya Shaastra as having been brought in to the present form around circa 400 BC.