The difference in lengths of a mean solar day and a side real day is about?
Solar time is time measured with respect to the Sun's apparent motion in the sky. The clocks we use for civil timekeeping are based on this motion. Of course, the apparent motion of the Sun across the sky is actually caused by the rotation of the Earth. So, our clocks measure the length of time required for the Earth to rotate once with respect to the Sun. From our perspective, the Sun revolves around the Earth every 24 hours. This period is known as a solar day.
Sidereal time is time measured with respect to the apparent motion of the 'fixed' stars in the sky due to the Earth's rotation. While the Earth is rotating on its axis it is also moving along its orbit around the Sun. Over the course of a day the Earth moves about one degree along its orbit (360 degrees in a full orbit divided by 365.25 days in a year is about one degree). Therefore, from our perspective, the Sun moves about one degree from west to east with respect to the 'fixed' stars.
All of this means that according to our clocks, which are based on solar time, a given star will rise or set about four minutes earlier each day (the Earth rotates 15 degrees in one hour, i.e. 360/24, so one degree of rotation is equivalent to about four minutes of time). For example, a star that rises at 9:00pm (21:00) tonight will rise at 8:56 pm (20:56) tomorrow and at 8:52pm (20:52) the next night. One month later that star will rise or set two hours earlier. In other words, from our perspective, the stars revolve around the Earth in only 23 hours and 56 minutes. This period is known as a sidereal day.