21st Century
Years of the Gregorian calendar, which is currently in use, are counted from AD 1. Thus, the 1st century comprised the years AD 1 through AD 100. The second century started with AD 101 and continued through AD 200. By extrapolation, the 20th century comprises the years AD 1901-2000. Therefore, the 21st century will begin with 1 January 2001 and continue through 31 December 2100.
Third Millennium
Similarly, the 1st millennium comprised the years AD 1-1000. The 2nd millennium comprises the years AD 1001-2000. The 3rd millennium will begin with AD 2001 and continue through AD 3000.