According to Wikipedia: 2000 = "the two thousands" http://en.wikipedia.org/wiki/2000s_(decade) 2010 = "the twenty-tens", "the two thousand tens" or "the two thousand and tens" http://en.wikipedia.org/wiki/2010s To avoid confusion with the whole century, it would be possible to say "the early two thousands" to refer to 2000-2010. 2000 to 2100 is also called the twenty-first century.

2013 would be in "the tens". 2001 would be in "the two thousands". The first decade of a particular century is just referred to as the century itself, like "the nineteen hundreds". Of course, a full century is also referred to in this manner, so it becomes ambiguous whether you're dealing with the first decade of a century or the century itself. When that happens, you'll need to use the context to figure it out.

Here in the UK, the 2000s are somtimes refered to as "the naughties" by newspapers and such.

thank you all for your concern

Those years are indeed difficult to describe eloquently (especially in writing). You might try - "At/during the turn of this last (or 21st) century..." or "At/during the turn of the new (or third) millennium..." or "During the early years of the 21st century..." which would describe the span of years leading up to 2020.