Every calendar in the world solves the same arithmetic problem, and there are only three solutions. A lunation is about 29.53 days and a tropical year about 365.2422 days, and the second is not a whole multiple of the first. A calendar can therefore follow the Moon and let its year drift against the sun, follow the sun and let its months lose contact with the Moon, or follow both and insert an extra month at intervals to keep them together.
The Hindu calendars take the third path, and they add something the others do not have. A Panchanga is not primarily a list of dates but a table of five limbs, and only one of those five is the lunar day. The others are the weekday, the nakshatra the Moon occupies, the yoga computed from the combined longitudes of Sun and Moon, and the karana, which is half a tithi. A day is therefore described by five simultaneous positions rather than by a single number.
The tithi itself is unlike a day in any other system. It is the time the Moon takes to gain twelve degrees of longitude on the Sun, so it varies in length with the actual motion of both bodies, and it does not align with the civil day. A tithi can be skipped or repeated in a given civil month, which is a feature of the definition and not an irregularity.
There is no single Hindu calendar. Regional systems differ in whether the month begins at the new or the full moon, in whether the year is reckoned from a solar or a lunar starting point, and in which era they count from, so two Panchangas printed in different parts of India can give different month names for the same day.