The Hindu calendar is extremely complex in that the dates do not always change at the same hour, sometimes days are skipped and months are added. But despite this it is scientifically precise in that one can predict days, dates months forward and backwards from now till eternity, just like one can do in the Gregorian calendar. The Hindu calendar is essentially lunar, but has some attributes of the solar cycle as well. Hence it is often called “luni-solar”.
A year consists of twelve months each month corresponding to a complete lunar cycle. As measured against our 24 hour day, the length of a lunar day varies, averaging a little less than 24 hours. This is the reason the dates do not change at the same hour. The lunar month is about 29.5 days. Hence the year has 354 days, 11 days short of the Gregorian year. As compared to the Gregorian year dates keep getting pushed back by 11 days every year. This is a major problem, because the seasons play an important role in Hindu life, as they do elsewhere. And seasons depend on the solar cycle. Hence about every three years an additional month is added to the calendar to reconcile it with the solar cycle. The year is divided into two halves – one when the sun is directly overhead in the northern hemisphere (known as uttarayana) and the other when the sun is directly overhead in the southern hemisphere (daksinayana). It is also divided into six seasons.
A month consists of two parts depending on whether the moon is waning or waxing. Each part consists of fifteen days. Since the months is 29.5 days and not 30 days, at the required times a day is skipped. The numbering of the days does not go from 1 to 30 but from 1 to 15. It starts with the first day of the waning moon (dark fortnight or krishna paksha) and on the fifteenth day of the waning moon is the new moon day. The date then becomes the first day of the waxing moon (bright fortnight or shukla paksha) and on the fifteenth day is the full moon day.
Not everyone can read the calendar because of the peculiarities explained and this is its major disadvantage. However it is used largely for religious functions and for preparing horoscopes so one need not get involved with the intricacies.
All ancient societies had lunar calendars. Undoubtedly the common folk would have found it easier to measure the progress of the days by looking at the phases of the moon rather than the position of the sun. Remember there were no time pieces or printed calendars then.