There are people going to celebrate the end of the century with a party on 31/12/1999 and others on 31/12/2000. Which is correct?

Probably a lot are going to parties on both! In fact any period of 100 years is a century. 1897 to 1996 is a century, and a new century starts on the 1st of January every year. Similarly 1997 to 2996 is a new millennium. But these are not useful, and people have named some decades, centuries and millennia. Strangely, they have named the centuries and millennia differently from the decades.

Despite changes in the past our calendar is now calculated from a specific date when something may [or may not have happened]. That date is the 1st of January, year 1: there was no year nought! So the "first" century was years 1 to 100, the second 101 to 200, ... the twentieth century is 1901 to 2000 and the twenty-first century will be 2001 to 2100. Similarly the first millennium was 1 to 1000, the second is 1001 to 2000, and the third will be 2001 to 3000.

However, decades are different. We refer to the nineties [short for nineteen-nineties]: these are 1990 to 1999. If we used the equivalent names to centuries, then we would call the current decade the "two hundredth" decade, which is 1991 to 2000. Remember, there was no year nought.