The editors of Scientific American offer this explanation:
Years in the most popular calendar used today, the Gregorian calendar, are counted from the year A.D.1. There was no year 0. Before A.D.1 came the year B.C.1. Thus, the first century ran for 100 years from A.D.1 until the end of A.D. 100; the first millennium, from A.D.1 until the end of A.D. 1000; and so the current millennium will not end until December 31, A.D. 2000.
A 6th century scholar, Dionysius Exiguus (Dennis the Short), established the Gregorian calendar in A.D. 532 by fixing A.D. (Anno Domini)1 as the time of Jesus Christ's birth. In Dionysius' time, the notion of counting from 0 had not yet been introduced to Europe from the Middle East. Jesus Christ was more likely born in B.C. 6, but Dionysius' system has held firm throughout the years. There are, however, some 40 other calendar systems in use, all of which are in different years that change on different dates.
Officially, the new millennium will begin at zero hour, Greenwich Mean Time (GMT), also referred to as Coordinated Universal Time (UTC), on January 1, 2001, according to rules adopted at an international conference held in October 1884. But that same conference also decided that this reckoning "shall not interfere with the use of local or other standard time where desirable." In other words, everyone east of Greenwich will not postpone their parties past midnight local time, and everyone west won't celebrate early.
The year 2000 is special--even though it isn't the start of the 21st century--because it is a leap year. Julius Caesar devised the leap year to correct for the fact that the earth circles the sun in 352.24219 days. Because this is not a whole number, the months of the year would slowly fall out of sync with the seasons. A fairly precise correction to the Gregorian calendar debuted in 1582, and stated that a century year will only be a leap year if it is evenly divisible by 400--which is true for Y2K.
Image: COMMANDER JOHN BORTNIAK, NOAA Corps
Frank Morgan, the Meenan Third Century Professor of Mathematics at Williams College, gives the following answer, adapted from his upcoming Math Chat Book, which is based on his Math Chat TV show and column, both available at the Mathematical Association of America's Web site:
The inexorable mathematical logic is that the official calendar millennium does not start until the year 2001. The first 2000 years end with the year 2000, and the next thousand start with 2001, the first year of the third millennium. Imagine a vast army of soldiers, with 1,000 men in each row. In the first row are soldiers 1 to 1,000, in the second, 1,001 to 2,000, and in the third, 2,001 to 3,000. The third row starts with soldier 2,001. Or suppose you work 1,000 hours a year. The first year, you work hours 1 to 1,000, the second year hours 1,001 to 2,000, and the third year begins with your 2,001st hour of work. So we should definitely celebrate the official calendar millennium on January 1, 2001.
But there is another millennium to celebrate: the millennium of the 2000s, the years that begin with a 2. This change will affect every check we write, every letter we date. It is exciting to see all the digits roll over for the first time since the year advanced from 999 to 1000, when Ethelred II was king of England; as exciting as seeing the odometer in my Ferrari roll over from 99,999 to 100,000 or seeing the whole Senate roll over to 100 new Senators (which probably never will happen, but then again, I don't really have a Ferrari either). Of course, this change in date is what causes the Y2K problems with computers, which will interpret '00 as 1900 instead of 2000. So maybe it's safer to wait until the official 2001 to celebrate.