How & why does a day measured by the sun differ from a day measured by the stars?

Answer 1

A sidereal day is shorter than a solar day because although the Earth is rotating with a 24hr period, it is also orbiting the Sun with a period of 365.25 days.

This means that every 365.25 days, you get a free rotation, so the sidereal day is 1/365.25 shorter than a solar day
i.e. 364.25/365.25 * 24hrs

Answer 2

The difference is due to the fact that the earth is in orbit about the sun. While the other stars are relatively fixed with respect to Earth.

During the time the Earth rotates on its axis relative to the distant stars (one *sidereal* rotation), it also moves laterally relative to the sun as it revolves around it. So the Earth has to turn just a bit further until it presents the same face to the sun as before. This is the normal "solar day" we experience, that lasts from one noon to the next.

So for Earth:
One sidereal day = 23 hours, 56 minutes, 4.091 seconds
One solar day = 24 hours

Answer 3

Answers above are good. But the question is even more complex because the Earth has different types of years. The types of years: 360 degrees around the sun, around the sun from one equinox to the next, around the sun from one perihelion to the next, around the sun from one orientation towards the "fixed" stars to the next.
Check out http://en.wikipedia.org/wiki/Year for other types of years.