Officially, the third millenium begins on Jan.1, 2000.
Historically, the chronology is more complicated:
The Christian Era was invented by Dionysius Exiguus in 525. The new chronology was a by-product of the dispute that had troubled the churches as to the proper method of calculating Easter. Many churches followed a method introduced by the bishop of Alexandria, but some Western churches followed other systems.
In 525, at the request of Pope St. John I, Dionysius prepared a modified Alexandrian computation. In the course of his new calculations Dionysius decided to discard the Alexandrian era of emperor Diocletian, which reckoned from AD 284, because he 'did not wish to perpetuate the name of the Great Persecutor of Christians, but rather to number the years from the Incarnation of Our Lord Jesus Christ'.
Dionysius wrongly dated the birth of Christ as 753 A.U.C. (Roman Era), but the Gospels state that Christ was born under Herod the Great who died in 750 A.U.C. The use of the Christian Era spread through the employment of Dionysius' new Easter tables and was later popularized by the English scholar Bede.
Because Dionysius had omitted to place a year zero between 1 BC (before Christ) and 1 AD (Anno Domini = the year of the Lord) there is a lingering dispute among chronologists: Should the proper millenium start in AD 2000, or in 2001 ? Mathematically, the correct answer would be 2001 since we should add 2000 years to the year 1. Traditionally, we settle for the year 2000, because it is a round number, although that date is obviously false.