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One Earth year was equal to 365.2425 Earth days in the Gregorian calendar. To compensate for the fraction of a day, a leap day was added to every year whose number was divisible by four, unless it was a century, unless it was divisible by 400. These leap years consisted of adding an extra day to the month of February. Instead of the usual 28 days, there would be 29.
Scientists usually used a Julian year of 365.25 days for measurements and scientific comparisons.
The mean solar year was 365.242190419 days in 2000 AD, and will be shorter still by the 23rd century.
A month was usually the amount of time it took for a moon to orbit its planet. This was usually a portion of a year, and a large number of days (in which case it may be broken down into weeks).
On planets without moons, a month could either A) not exist, B) be equal to a season, or C) be a fractional division of a season. For example, Vulcan "has no moon", so it is unknown what portion of a year is represented by the "month" of Tasmeen. The novelization of The Motion Picture could be read to imply that Vulcan's "months" are whole seasons.
On Earth a month was originally the length of the lunar cycle (29.53 days). Most calendars at some point made the month one twelfth of a solar year (30.44 days). In Earth's most common calendar, the months are either 30 or 31 days long, with one shorter month (February) having 28 or 29 days depending on the year.
A second was a portion of a minute, this could be a decimal hundredth of a minute, or some other fractional portion of a minute.
On Earth a second was 1/60 of a minute, and was usually divided up decimally.
The second is now precisely defined based on the atomic behavior of cesium, and the lengths of minutes, hours, and days are now defined in these seconds. This means that these other times are no longer exactly the same as their solar equivalents based on actual planetary motion.
In the 23rd century, stardates were not directly related to Earth's calendar. Beginning in 2323, stardates were changed to be 1,000 per Earth year.
The measurement of 1,000 stardate units per year was never explicitly stated on-screen, but can be inferred from numerous conversations. There are some apparent errors, or variations, however, in stardate calculation. One example is Annika Hansen's birth date being given as stardate 25479, and the year being given as 2350. If that is counted back, it would give the beginning year of 24th century stardates as 2325, which contradicts the Star Trek Chronology.
It is unknown exactly how many days are equal to 1,000 stardate units in the 24th century; it could be 365.25, 365.2425, or just 365.