Mean solar time… what does it mean?
Mean, in this case, means average. This is a confusing idea to use on time. Normally, one takes averages over time… travelling 10 miles in 1 hour means an average speed (averaged over time) of 10 miles per hour. How do you take an average of time?
The clue is in the word solar. Solar time is not a regular, clockwork kind of time. Near perihelion, the sun travels faster across the celestial sphere than at aphelion. Near the solstices, the sun travels faster (in the East-West direction, which is the only component of the sun's position used by most sundials) across the celestial sphere than near equinoxes. So, solar noon (noon as read by a sundial) on one day to solar noon on the next day (called a solar day) is sometimes more than 24 hours, and sometimes less.
A mean solar day is the average length of a solar day (as measured by a regular timekeeper). One year of days is enough samples from which to take a good average (in fact, only multiples of years should be used for taking such averages, so the net effects from the axial inclination and orbital eccentricity are close to zero).
Mean solar noon is the average time of mean solar day at which solar noon occurs. In other words, mean solar noon is a time chosen such that if someone were to make a list, every day marking whether the actual solar noon occurred before or after mean solar noon, and recording how many minutes difference there was between the two, after a year the total of the 'before' time differences is the same value as the total of the 'after' time differences.
Mean solar noon is the anchor of mean solar time. 3pm mean solar time is 3 hours after mean solar noon. All other mean solar times are figured similarly.
Of course, the most significant mean solar time is Greenwich Mean Time. This is the internationally recognised standard time, degrees longitude (the Prime Meridian) is defined to run through the Royal Greenwich Observatory so that there is a clear and direct relationship between local time and longitude (for every degree longitude difference from Greenwich, the mean sundial time differs by 4 minutes from GMT).