Post 365 seems a fitting place to discuss the measuring of time.
The average tropical yearTropical year = time between consecutive winter solstices lasts for 365.24219 SI days. 2012 exceeded this by 3 minutes and 35 seconds; 2011 fell short by almost 12 minutes. The average Gregorian year lasts for 365.2425 days, about 27 seconds too long. This is actually good for the long-term correctness of the Gregorian calender because on the whole years are getting longer over time.
The synodic monthSynodic month = time between consecutive new moons and varies between 29.2 and 29.9 days, but the average synodic month is know with great precision: currently it is about 29.53058888 SI days (29 days, 12 hours, 44 minutes, 2.879 seconds). The average month duration increases by approximately one second every 14 or 15 years.
The solar daySolar day = time between sun’s consecutive passes over the same longitude varies through the year in a manner than changes slightly each year. Solar days can be as short as 86,370 seconds near equinoxes and as long as 86,430 seconds near solstices. This variation has nothing to do with the “shortest day” which refers to the portion of the solar day during which the sun is above the horizon and always falls within a day of the winter solstice.
An SI day is exactly 86,400 seconds. A second is the time it takes a fully stationary caesium-133 atom to oscillate between the two hyperfine levels of its ground state 9,192,631,770 times. The hyperfine levels oscillate because of the interaction of the magnetic field generated by electron orbits, the magnetic field generated by the spin of protons and neutrons, and the distribution of protons and neutrons in the nucleus. The speed of hyperfine oscillations is fixed because the speed of light is fixed and nuclei and electron orbits have fixed sizes.
Rubidium-87 and hydrogen-1 can also be used to define time; rubidium cycles at approximately 6,834,682,610.904 Hz and hydrogen at 1,420,405,751.768 Hz. These are approximate where the caesium oscillation is exact only because the second is defined in terms of caesium oscillations; we know each frequency with similar precision.
Since the speed of light is a constant, we could define time based on distance but it turns out that we can measure time more accurately than distance so it actually goes the other way: a meter is defined as 1/299,792,458 of the distance light travels in a vacuum each second.
On the larger, less precise scale, there are lots of calenders in the world. Most are based on some combination of tropical years, solar days, and synodic months. Each calender has its own set of rules for getting days, months, and years to line up. Some, like the Julian and Gregorian, ignore synodic months and just try to sync up days and years. Others, like the Hijri calender, ignore solar years and just try to sync up days and synodic months. Some are based on observations (e.g., “the month of the barley harvest in Judea” or “the first day a crescent moon is observed after sunset”) and others on calculations (lead years, leap days, 13-month years, etc.) A few are really complicated by adding in other cycles; for example, in the Hebrew calender there are special changes to ensure Yom Kippur is not a Friday or Sunday and Hoshana Rabbah is not a Saturday.