The lunar month: synodic and sidereal
The synodic month, from new Moon to new Moon, is about 29.53 days. The sidereal month, the Moon's return to the same star, is about 27.32 days.
What it is
The difference comes from the Sun's motion: during the sidereal month the Sun moves about one sign's worth of distance, so the Moon must travel farther to catch it again.
How it is worked out or read
The tithi and the paksha depend on the synodic month, while the nakshatra depends on the sidereal month. Both are used in the panchang.
Where it appears on this site
See lunar month systems and tithi.
Two kinds of month
The Moon completes a circuit against the stars in about 27.32 days. This is the sidereal month, and it is the period in which the Moon passes through all twenty-seven nakshatras, at a rate of a little more than one nakshatra a day. But the Moon returns to the same phase, from new moon to new moon, only after about 29.53 days. This longer period is the synodic month, and it is the month that the panchang counts in tithis.
The two differ because the Sun is also moving. While the Moon goes once around the zodiac, the Sun advances by about 27 degrees, and the Moon needs roughly two more days to catch up with it. The synodic month is therefore the one that governs the phases, the festivals and the lunar calendar, while the sidereal month governs the nakshatra cycle.
Why the month varies
The figure of 29.53 days is an average. The Moon's orbit is an ellipse, and the Earth's orbit around the Sun is one too, so the actual time from one new moon to the next can be several hours shorter or longer than the mean. The variation can reach more than half a day in either direction. This is one reason why tithis are not equal in length: a single tithi can last anywhere from about 19 to about 26 hours, depending on how fast the Moon is moving relative to the Sun at the time.
Panchang makers who use the true positions of the Sun and Moon capture this variation automatically. Older methods that used mean motions would give months of equal length and so place new moons and tithi changes at slightly wrong moments.
Twelve months and the solar year
Twelve synodic months add up to about 354.37 days, roughly eleven days short of the solar year. If the lunar calendar ran on its own, its months would slip through the seasons, as happens in some purely lunar calendars. The Indian luni-solar calendar corrects this by inserting an extra month, the adhik maas, roughly once every 32 to 33 months, in a lunar month that contains no sankranti. This keeps the festivals in their seasons over the long run.
Amanta and purnimanta months
The length of a month is the same whichever way it is counted, but its starting point differs by region. In the amanta system, used widely in the south and west, the month ends at the new moon. In the purnimanta system, used widely in the north, it ends at the full moon. The two systems give the same tithi on the same day but can give different month names for the dark fortnight. A reader comparing almanacs from different regions should check which convention each uses before concluding that the dates disagree.