Solar time is based on the idea that, when the sun reaches its highest point in the sky, it is noon. Apparent solar time is based on the apparent solar day, which is the interval between two successive returns of the sun to the local meridian. Solar time can be measured by a sundial.
The length of a solar day varies throughout the year. This is because the Earth's orbit is an ellipse, and not a circle, and the Earth moves faster when it is nearest the Sun and slower when it is farthest from the sun. (see Kepler's laws of planetary motion) Because of this, apparent solar days are shorter in March and September than they are in June or December. (The amount of daylight also varies because of the 23.5º tilt of the Earth's axis. (see Tropical year)
Mean solar time is based on a fictional mean sun which travels at a constant rate throughout the year. The length of a mean solar day is a constant 24 hours throughout the year although, as noted above, the amount of daylight varies.
The difference between apparent solar time and mean solar time, which is sometimes as great as 15 minutes, is called the equation of time.
The above is the standard explanation of apparent solar time and mean solar time found on most encyclopaedias, and it is "good enough" for most purposes.
However, it contains a flaw. It is in the pass from " Apparent solar time is based on the apparent solar day, which is the interval between two successive returns of the sun to the local meridian. The length of a solar day varies throughout the year. " to " Mean solar time is based on a fictional mean sun which travels at a constant rate throughout the year. The length of a solar day does not vary. "
The last sentence makes as much sense as saying " mean time is based on a fictional mean ant which travels at a constant rate throughout the year around a fictional circle ". Of course, if we had an ACTUAL ant which moved at a constant rate, we could check the length of the solar days against it, make observations of the sun, and say things like "from yesterdays' noon to today's noon 23 hours 59 minutes 15 seconds of constant-rate ant time have elapsed". But a FICTIONAL ant won't do the trick, because we can't measure its position in the fictional circle. So mean solar time is not based on it.
Time can only be measured by counting the repetitions of a phenomenon, be it natural like an astronomic observation or artifficial like the ticks of a clock. If we had a reliable enough clock, we could use it to count the number of clock-ticks from a noon to the next and say that the length of the solar days, ACCORDING TO THE CLOCK, varies throughout the year. Of course, if all of our clocks are made with XVI century technology so we can't rely on them, then we would be safer saying that, according to the Sun, clock rates vary throughout the year.
Up to the middle of the XXth century, mean time was based on observations of the Sun which were used to keep right the "master" clocks on which civil time was based.
In fact mean solar time was not based in the orbital motion of the sun around the Earth, on the apparent motion of stars around the Earth, on clocks, or on anything else. When it was used, mean solar time was based exclusively on observations of the apparent motion of the Sun around Earth. To "measure" mean solar time, first the apparent solar time (the position of the Sun) was observed, and then a mathematical correction was added to or substracted from it to get "mean" time. This correction was purely theoretical and based on theories of the motion of the Earth around the Sun &c, and the resulting "mean" time was found experimentally to agree with the best mechanical clocks up to the beginning of the XXth century.
Nowadays mean solar time is no longer used to keep clocks right, except in Saudi Arabia. Current atomic clocks are widely believed to be more reliable than the rotation of Earth, and the Universal Time scale (TU) our clocks are based on is based on them.
Universal time is never more than one second away from Greenwich's mean solar time. According to the atomic clocks, the rotation of the Earth gradually becomes slower, so leap seconds must be inserted occasionally in the TU scale to keep Greenwich mean noons near 12:00:00 TU.