For years between 2000 and 2010, it is common to say the whole number:
2000 = two thousand
2001 = two thousand one
2002 = two thousand two
...
2009 = two thousand nine
Beyond the initial ten, we tend to say them in pairs:
2010 = twenty-ten
2011 = twenty-eleven
2012 = twenty-twelve
2020 = twenty-twenty
Sometimes we even say some of the first ten in pairs (though never for 2000 and this is pretty colloquial):
2001 = twenty o one
2002 = twenty o two
2008 = twenty o eight
2009 = twenty o nine
It is by no means wrong to say the full number and it is encouraged in a formal setting and in writing:
2010 = two thousand ten
2011 = two thousand eleven
2020 = two thousand twenty
We also don't tend to say "year" unless needed:
It's twenty-nineteen!
I was born in nineteen ninety one.
But you can:
I was born in the year nineteen ninety one
This year is twenty-nineteen!
Note that you can't just add "year" in front of the number and expect it to be correct. As in the two preceding example sentences, you need to add either "the year" or "this year is" etc.