Julian Days are a way of reckoning the current date by a simple count of the number of days that have passed since some remote, arbitrary date. This number of days is called the Julian Day, abbreviated as JD. The starting point, JD=0, is January 1, 4713 BC (or -4712 January 1, since there was no year '0'). Julian Days are very useful because they make it easy to determine the number of days between two events by simply subtracting their Julian Day numbers. Such a calculation is difficult for the standard (Gregorian) calendar, because days are grouped into months, which contain a variable number of days, and there is the added complication of Leap Years.
Converting from the standard (Gregorian) calendar to Julian Days and vice versa is best left to a special program written to do this, such as the KStars Astrocalculator. However, for those interested, here is a simple example of a Gregorian to Julian day converter:
JD = D - 32075 + 1461*( Y + 4800 + ( M - 14 ) / 12 ) / 4 + 367*( M - 2 - ( M - 14 ) / 12 * 12 ) / 12 - 3*( ( Y + 4900 + ( M - 14 ) / 12 ) / 100 ) / 4
where D is the day (1-31), M is the Month (1-12), and Y is the year (1801-2099). Note that this formula only works for dates between 1801 and 2099. More remote dates require a more complicated transformation.
An example Julian Day is: JD 2440588, which corresponds to 1 Jan, 1970.
Julian Days can also be used to tell time; the time of day is expressed as a fraction of a full day, with 12:00 noon (not midnight) as the zero point. So, 3:00 pm on 1 Jan 1970 is JD 2440588.125 (since 3:00 pm is 3 hours since noon, and 3/24 = 0.125 day). Note that the Julian Day is always determined from Universal Time, not Local Time.
Astronomers use certain Julian Day values as important reference points, called Epochs. One widely-used epoch is called J2000; it is the Julian Day for 1 Jan, 2000 at 12:00 noon = JD 2451545.0.
Much more information on Julian Days is available on the internet. A good starting point is the U.S. Naval Observatory. If that site is not available when you read this, try searching for “Julian Day” with your favorite search engine.