Those living in early twentieth century called the years 1900-1909 "The Aughts" .
This Wikipedia article provides some more context.
english.stackexchange.com/users/527/Portman
I've heard "the noughties" (easy transition from "nineties") and "the two-thousands" (literal interpretation of "2000s") for the decade just gone.
I think this is something that doesn't have a real answer as it appears to change with each century and is different depending on your location.
Wikipedia's entry for the 2000s echoes this inability for people to come up with a single name for this decade
let alone all "first decades of a century".
english.stackexchange.com/users/250/soutarm
In my circle I have usually heard the previous decade referred to as "the two-thousands".
As to whether it might be called that in 30 years or so, it's hard to tell.
I agree that "the tens" sounds weird, but isn't that because you'd say "the teens" instead?
english.stackexchange.com/users/449/Kosmonaut
The "Noughties" became a fairly standard term pretty quickly, at least in the UK.
But I've still yet to hear anyone give a good term for the decade from 2010-2019.
If I had to guess what term is most likely to emerge for 2010-2019, I'd say perhaps "The Teens", but I suspect actually no term will catch on until the Twenty-Twenties.
english.stackexchange.com/users/772/AlexC
I can only think of "the single digits" to call the first nine years of the new millennium.
english.stackexchange.com/users/38134/Geoff L
I usually hear the decade of 2000 through 2010 referred to as the two thousands.
But the year 2000 is really the last year of the 1990s.
For consistency the decade 2001-2010 should be referred to as the 20 zeroes and this decade 2011-2020 is the 20 tens.
Like every other decade we just say the number with the zero and add an s.
english.stackexchange.com/users/91476/Pooper Magic
I will offer the correct answer.
Quite simply, for the two decades 2000-2010, and also 2010-2020 (the current decade), in fact there is no popular slang term .
There were attempts to popularise "naughties" but it never caught on.
Regarding back in 1900-1920.
I have not been able to find any clear reference on the issue.
english.stackexchange.com/users/8286/Fattie
The first ten years of any century could accurately be referred to as the 'singular', or 'unit', years, as opposed to those of double digits that follow.
That would be a designation consistent with the linear principles of fundamental mathematics, no?
english.stackexchange.com/users/95470/Gregory Bryant
Fot the current century, I would propose:
two thousand zeroes
My reason being, I think most people would understand what you meant.
(because of its recentness)However Im not so sure it would work for the first decade of a century in general.
i.e. in the fifteen zeroes davinci focused his attention on developing military machines for his patrons.
Its a little more ambiguous, but maybe understandable...
english.stackexchange.com/users/171618/the_velour_fog
FWIW, this period has been called the 'Zeroes' by a Dutch radio station : The first decennium of this century is described as the 'Zeroes'.
It concerns the first ten years of this millennium.
english.stackexchange.com/users/116151/Glorfindel
80s: the "Eighties"    90s: the "Nineties"    00s: the ???   For that matter, what is the second decade called? The "tens" just doesn't sound right.
What is the name of the first decade in a century?
english.stackexchange.com
<word-choice><numbers><dates>
english.stackexchange.com/questions/997
english.stackexchange.com/users/460/Chris Noe
There are a number of different potential names for the first decade in a century. Those living in the early twentieth century call that decade "The Aughts", whilst the terms "the noughties" and "the two-thousands" have also been used for the first decade of this century. There is no fixed term - both "noughties" and "two-thousands" have been used for 2000-2009 and no term exists for the years 2010-2010.
There are a number of different potential names for the first decade in a century. Those living in the early twentieth century call that decade "The Aughts", whilst the terms "the noughties" and "the two-thousands" have also been used for the first decade of this century. This appears to change with each century and is different depending on your location.  The "Noughties" became a fairly standard term pretty quickly, at least in the UK. And it was referred to as the 'Zeroes' by a  Dutch radio station. "The two-thousands" is another option, or "the single digits". Quite simply, for the two decades 2000-2010, and also 2010-2020,  in fact  there is no popular slang term .  
"The noughties" and "the two-thousands" are commonly used for 2000-2009 and a dutch radio station uses the term "the zeroes". One suggestion for 2010-2019 is "the teens". There is currently no agreed term for the first or second decade of the century. However, various names have been suggested for the first decade, including "the naughties", "the single digits" or "the two thousands". Additionally, a suggested term for the second decade of the century is "the teens".
The years 1900-1909 were called the "aughts" There is no fixed term - both "noughties" and "two-thousands" have been used. The "noughties" is common in the UK at least, but there is no term for the years 2010-2019. The decade of 2000 through 2010 is referred to as the two thousands. There is no set term.  The first ten years of any century could accurately be referred to as the 'singular', or 'unit' years.
The "Noughties" became a fairly standard term pretty quickly, at least in the UK. And it was referred to as the 'Zeroes' by a  Dutch radio station. This appears to change with each century and is different depending on your location.   "The two-thousands" is another option, or "the single digits". Quite simply, for the two decades 2000-2010, and also 2010-2020,  in fact  there is no popular slang term .  
There are a number of terms that are in use to name the first decade of the 21st century, including "the naughties", "the two thousands" or "the single digits". There is currently no agreed upon term for the first or second decade of the century. One suggested name for the second decade of the century is "the teens".