A millennium is 1,000 years but confusingly, the next Millennium doesn't start in 2000 - the 21st Century and the Third Millennium start in 2001. Why? Because in the 6th century AD, a monk called Dionysius Exiguus who was working out the dates decided to reset the system of counting years to honour the birth of Christ, numbering each year Anni Domini Nostri Jesu Christi (the Year of Our Lord Jesus Christ). He believed that Christ was born on the 25th of December of the year preceding the start of the year 1 AD (although modern research suggests that Christ was probably born in 6BC and certainly by 4BC when Herod died) and because he'd never heard of zero, he started with 1 AD. Two thousand years later is 2001 - and we officially end the year on December 31 Common Era (CE) rather than December 25th... Of course it's only the Christian calendar that thinks it's the Third Millennium. On 1 January 2001, it's 7509 in the Byzantine calendar, the Chinese calendar says 4637, Indian (Saka) 1922, Islamic (Hegira) 1421 and Jewish (A.M.) 5761.
And whether you're celebrating 2000 or 2001, when does it start? The Royal Observatory at Greenwich says that because the meridian at Greenwich is the prime meridian for "all the world" then the next Millennium starts at zero hours Greenwich time. And where is the first sunrise of the Millennium? At midnight Greenwich time on 1 January 2001 the Sun will be rising along a line running from Eastern Russia, through China out over the Bay of Bengal and down towards the Antarctic circle; the first UK sunrise will be at South Foreland near Dover and if you want to you could then fly 1,500 miles west to Samoa and see the last sunset of the old Millennium, 14 hours and 14 minutes later. Remember though - because the Earth's atmosphere bends light, you can see the disk of the Sun in the sky before the real Sun has risen.