This topic has had loads of debate and wikipedia explains it well. Found this nice thread on Reddit. A decade, century or millennia can be any arbitrary period of 10, 100 or 1000 years. For example 1450 to 1549 is a century. However, if you want to refer to a time period relative to another, the counting matters and this causes different points of view.
So you can legitimately claim a millennium started in the year 2000 and many did. That's fine. However from a math point of view, the 3rd millenium didn’t start until 2001. Likewise, you can claim that 2000 was the beginning of a new century however the 21st century began in 2001 (the first century was 1 to 100, 2nd century 101 to 200, etc).
But decades aren’t usually referred to relative to other decades. We just say 60’s, 70’s and 80’s. So it makes sense to say 2010 is the start of a new decade. But if you want to be really pedantic, the 202nd decade of the common era starts in 2011.