The beginning of the twentieth century was almost universally hailed on January 1, 1901, rather than January 1, 1900, by the general public and by newspapers with banner headlines. However, this time around, most people celebrated the new millennium in January 2000; there were no banner headlines in January 2001. As Steven Gould has pointed out, this change reflects the ascendance, during the last hundred years, of popular culture over a pedantic elite.
Popular equates here with common sense: the transition from 1999 to 2000 was the obvious one. Pedants base their case on the historical accident which left no year zero between years 1 BCE and 1 CE. The hundred-year period beginning with the year 1 and continuing through the year 100 has thus been called the "first century" --- even though there had already been dozens of historical centuries.
In fact, there is a new century beginning whenever we feel like counting, lasting then for a hundred years. The question is which centuries we choose to name. "The 1900s" unambiguously means the century beginning with 1900 and ending with 1999. It is confusing enough that this is called not the 19th but the 20th century; why add to the confusion by insisting that the 20th century was delayed by one year?
The fact that people once had a different view of when such named centuries started shouldn't concern us now; years used to start in March, but nobody claims the new millennium thus starts in March.
Buildings rarely have a room number 100, but when they do, it is without exception on the same floor as rooms 101 and 117. Furthermore, this floor is the one called the "first floor", whether that means the ground floor as in the US, or the one above as in Europe. It would be most sensible to call the century associated with the number 19 (that is, the 1900s) the 19th century; unfortunately, we are stuck with our less logical convention.
The numbering of years in our Common Era was set in 525 CE by the monk Dionysius Exiguus, who had been asked to work out the dates of future Easters. He knew roughly how many years it had been since the birth of Jesus (529 according to modern scholarship), but he evidently chose the exact figure 525 to avoid a remainder term in his Easter formula.
Even several centuries later when dates BCE came into use, western mathematics did not have the concept of zero. The common system thus skips year zero, though modern astronomers have sensibly adopted a convention that includes year zero (and thus disagrees by one year with historians' dates BCE ).
Other dating systems have not faced the question of a year zero, because they did not attempt to rename prior dates. Even the French revolutionaries, with their zeal for logic in the calendar, and their certain knowledge of negative numbers, used the Common Era for years before the revolution.
The disagreement about the start of the millennium, coming from the lack of a year zero, is an example of a "fence-post error", which leads certain counts to be off by one. Four lengths of fence, for instance, will require five posts. Other examples abound: Musical intervals are named inclusively, so that a third plus a fourth is not a seventh but a sixth. What is called a "fortnight" in English is a "quinzaine" in French, and might be sold as a 15-day/14-night vacation.
The New York marathon evidently took advantage of this ambiguity to celebrate their "25th" race one year, and their "25th anniversary" race the next year (25 years after their first race). On the other hand, the TV show "Saturday Night Live" jumped the gun twice, celebrating their 15th and 25th "anniversary" seasons each one year early.
We were right to celebrate the new millennium in January 2000. Although there is a period of 1000 years that started in January 2001, it is not one worth noting. There was no Y2k+1 bug; there was no need to reprint blank checks; nobody suddenly felt that dates look weird.
Some interesting linguistic questions do remain open. Does "the 2000s" refer to a decade, a century or a millennium (and how is it pronounced)? Can we talk about the "twenty-hundreds"? What will we call this current decade, which comes a hundred years after the Naughty Aughties?
Compare the views of Helmer Aslaksen and Sean Oberle.