This millennium (and century) officially began on: 1 January 2001 GMT
The simple explanation is that our Calendar System began at 1AD. There was no year 0. The calendar we use today goes from 1BC to 1AD.
The calendar base that we use today was not conceived until the year 525 AD. At that time the Roman Calendar was still being used which was based on the founding of the city of Rome on April 22, 753 BC. The new calendar was conceived by a monk called Dionysius Exiguus. He proposed that Christ was born in the year of Rome 753 (but most historians agree that it should have been some years earlier).The Venerable Bede, an English monk, through his writings in the 8th century re-chronicled history. Up to that time people had been living, say, in the (Roman) year 1500 suddenly found they were living in 747 AD. (I.E. Don't believe anyone who finds a coin with a year in the first 6 centuries AD (they didn't exist for the people living in that time)!