The very first year was year 1. They didn't start counting dates as 1-1-00. The first day of the first year was 1-1-01. By that theory, we will soon be completing year #1,999, and just starting the 2,000th year.
I guess if you really get down to it, all anyone is celebrating is the fact that the calender was created ~2000 years ago. Since man has been around far longer, what's the point of the millenium?