The current decade, between the years 2000 and 2010. In English, (most often British English), "ought" can mean "zero", thus the "oughties". This is used in lieu of zero, because "zeroties" sounds awkward. Unlike with the "eighties", the "nineties" and so on, there is no one name for the current decade enjoying wide usage.
"I can't recall the exact year, but George Harrison died some time in the early oughties."