A millennium is any period of a thousand years. So you could say that the next millennium begins now. When people ask this question they are usually thinking of counting successive millennia of the Christian Era, using the A.D. (Anno Domini - year of our Lord) numbering scheme devised by Dionysius Exiguus (Dennis the Little) in the 6th century. The year before 1 A.D. was called 1 B.C. so the first thousand years A.D. started at the beginning of 1 A.D. and ended at the end of 1000 A.D., and the second thousand ends at the end of 2000 A.D., so the third millennium of the Christian Era began at the start of the year 2001 A.D.

The year 2000, the final year of the 20th century A.D., and the final year of the second millennium, was, of course, widely celebrated as marking the start of a new millennium.