A millennium is a period of one thousand consecutive years. When dating years, millennia are usually taken to begin in the years divisible by one thousand, or alternatively in the year after that, e.g., 2000 or 2001. The use of the odd year came from the Anno Domini system of religious dating, where the year 1 was supposedly the first "year of our lord" and thus the first millennium would be years 1 to 1000 and the second millennium would start 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 based on the Anno Domini system, the "true" new millennium would begin in 2001. Others argued that under the Gregorian Calendar centuries should be observed at the first numerical departure from the previous millennium (ie: we are in a new millennium when we stop being in the 1900s). Others argued that our entire system of dates is so arbitrary and meaningless that it is impossible to define the "true" date of a millennium.
Wikipedia has a page for each millennium: See millennia.
Millennium was also a suspenseful television series, produced by the creators of The X-Files and set during the run-up to the new millennium (it premiered in 1996).
It featured actor Lance Henriksen as investigator Frank Black.
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