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A millennium is a period of time equal to one thousand years. When dating years, each millennium begins the year after a year divisible by 1000, e.g. 1001, 2001. The use of the odd year is due to the fact that the Gregorian Calendar has no year zero, and thus the first millennium is years 1 to 1000 and the second millennium starts in 1001.
There was a popular debate leading up to the celebrations of the year 2000 as to whether 2000 was the true "new millennium". Some argued that the new millennium should begin when the first digit of the year changes; however, as there is no year zero in the Gregorian calendar, this would result in the first millennium being 999 years long. Others, such as Douglas Adams, argued that our entire system of dates is so arbitrary and meaningless that it is impossible to define the "true" date of a millennium.
BambooWeb has a page for each millennium: See millennia.
See also: Millenarianism