My father and I argued many times and could not convince each other. Eventually I asked Science Line.
The answer is:

" The 21st century will start from 2001.
There was no year zero in the calender, it weent from 1BC to 1AD.
Take an easier sample. When you were born you were zero years old.
Until your first birthday, you will still in your first year.
After you have your first birthday you're in your second year.
You don't start a new decade until the end of your tenth year - after your 11th birthday.
The same is true for the Earth.
So the next millennium won't start until Jan. 1st, 2001."

We both agreed on their explanation.
However, in practice, we will celebrate the New Millennium when Big Ben strikes 0.00 AM on Jan. 1st  2000. Because at that time the New Millennium is born, even he is "zero years old".

Science Line is a free public science information service.
Their URL:
http://www.campus.bt.com/CampusWorld/pub/ScienceNet/