A period of 100 years. Since there was no year 0, the years 1-100 AD correspond to the first century, 101-200 to the second, etc. The twenty-first century therefore began on January 1, 2001. The situation is summed up succinctly in the following editorial from the Times of London on December 26, 1799 (NPR Morning Edition), which discusses the issue of when the new century starts.

"We have uniformly rejected all letters and declined all discussion on the question of when the present century ends, as it is one of the most absurd that can engage the pubic attention, and we are astonished to find it has been the subject of so much dispute, since it appears plain the present century will not terminate until January the first, 1801, unless it can be made out that 99 are 100. It is a silly, childish discussion, and only exposes the want of brains in those who maintain a contrary opinion to that we have stated."

However, rather than wasting time arguing about whether 1900 or 1901, 2000 or 2001, etc. is the start of the "new" century we can simply note that the years 1900, 2000, etc. correspond to an odometer-like "rolling over" of the digits of the year, a very rare event which is clearly calendrically significant!