The solar noon is defined as the time of the highest position of the Sun in the sky and occurs when the Sun crosses the meridian at a given position. The length of the solar day is the time between two consecutive solar noons.
The mean length of the day, namely 24 hours, is a little less than the period of rotation of the Earth around its axis, since the Earth makes 366 rotations around its axis during a year of 365 days. If the axis of the Earth were vertical and the orbit of the Earth around the Sun circular, the mean length of the day would correspond to the time between two consecutive solar noons and therefore to the length of the solar day. For Greenwich’s meridian, the official noon is defined as the solar noon at the spring equinox, and then for the other days of the year by applying a period of 24 hours.
The solar noon oscillates during the year; it coincides with the official noon only on four days during the year. The equation of time is the difference between the solar time and the official time (mean solar time).
The fact that the equation of time shows oscillations with an amplitude of approximately 30 minutes is explained by two phenomena. The first one is the obliquity (tilt) of the Earth’s axis: if the orbit of the Earth around the Sun would be circular, the official noon would correspond to the solar noon at the equinoxes and at the solstices; it would be after the solar noon in fall and spring, and before the solar noon in summer and winter. The second ingredient is the eccentricity of Earth’s orbit around the Sun: when the Earth is closer to the Sun (during the winter of the Northern hemisphere), it has a higher angular velocity around the Sun, yielding longer solar days.