The shortest apparent solar day is on the fall equinox in mid-September. Data from 1998 says that it was 24 hours minus 21.3 seconds long on September 16th of that year. The longest day is the winter solstice in December. That was December 22nd in 1998 and was 24 hours plus 29.9 seconds long. These values change slightly every few years, and would be significantly different if you were to look over thousands of years.
There are two main causes to this variation in length. The first is the same reason we have seasons; that is, because of the tilt of the earth on it's axis. The second is the fact that Earth moves around the sun in an ellipse instead of a circle, causing Earth to be closer or further to the sun depending on the time of year.
