We can mark the progress of the year by measuring the changing length of days. At the solstices, the day reaches its longest and shortest lengths. At the equinoxes, the length of day and night are observed to be equal.
It is easy to determine the equinoxes and solstices, a simple stick (stuck in the ground to serve as gnomon) and careful observations will suffice. The ancients noticed that the number of days between equinoxes and solstices is not equal. The implication is that the sun's yearly journey around the ecliptic is not of uniform speed.
So, time as told by the sun (say, on a sundial) is not equivalent to time as told by the stars; it is imperfect in that it slows down and speeds up. This corresponds to the changing position of the sun in the sky if measured from the same exact time each day, producing the analemma.
Since time as told by the sun does not run at constant speed, it must be corrected for (on a sundial, for example) by consulting the equation of time. The equation of time corrects for the difference between time as told by the actual position of the sun and the mean sun.
The mean sun is an imaginary body that averages over the varying speeds of the actual sun. Time as told by the mean sun is equivalent to sidereal time.
In ancient astronomical models, for example, those of Ptolemy, the mean sun plays an important role. The planets undergo periodic motions which are clearly connected to the position of the sun. Some "clock" is needed to sync the planetary motions in any model, and this is the role played by the mean sun.
For example, one of the fundamental observable properties of planets is their period retrograde motions. Planets, like the sun, are observed to slowly change their position with respect to the zodiac throughout the year. Overall, motion against the zodiac is eastward. Occasionally, however, planets are observed to slow, stop, and before backward against this regular eastward motion. This backwards movement is called retrograde motion.
The "superior planets" (in geocentric terms, those which are farther from us than the sun; in heliocentric terms, those which are farther from the sun than us) are observed to undergo retrograde motion whenever they are "in opposition" (i.e. on the opposite side of the zodiac from the sun. In heliocentric terms, it is easy to see why this is the case: if the planet is in opposition, that means we are passing between it and the sun. Since we are closer to the sun, and thus move more quickly, the planet temporarily appears to move backward, just like a slower moving car when passed on the freeway.
In Ptolemy's model, the retrograde motions of superior planets are put "in sync" with the mean sun. Ptolemy stipulated that the radius of each superior planet's epicycle must stay parallel to the radius of the mean suns orbit.
Here, P is the planet, O is the earth, and the dotted circle with a bar on top is the mean sun. Υ indicates the direction of the vernal equinox.
Ptolemy's model also syncs "inferior planets" (in geocentric terms, closer to us than the sun; in heliocentric terms, closer to the sun than the earth). For the inferior planets, it is the line EK between the equant (the point with respect to which the motion of the point K is uniform) and the center of the planet's epicycle, K. Sun, Earth, and vernal equinox are notated as before.
Note that, it is not just the shift to a heliocentric system which undermines the importance of the mean sun. We can see this in Copernicus' model, which also refers all planetary motion to the mean sun.
It was not until Kepler's Astronomia Nova (1609) that the idea of abandoning the mean sun and referring all planetary motion to the position of the actual sun was hit upon. Kepler's solution was pure engineering. Through trial and error, he discovered that all three current astronomical hypotheses (Ptolemaic, Copernican, and Tychonican (in which the earth is stationary, but all other planets orbit the sun, which orbits the earth) could be improved and made equally empirically adequate. Most famous here is Kepler's decision to abandon circles in favor of the ellipse. More significant for improving empirical adequacy, however, was his abandonment of the mean sun in favor of the actual sun.
It would take Newton, however, to provide a motivation for this decision. Unless one realizes that the cause of planetary motion is in some sense the sun, there is no natural motivation for using the actual rather than mean sun. If all one wants is a clock, the mean sun is clearly superior. Only once a causal story of the planetary motion was provided could the success of Kepler's pragmatic adjustment be explained.