TL;DR: The answer to what is Gjirokasër that it is an old Ottoman, mainly 17th-century, magical hillside town with 600 white and shale topped houses and a spectacular view. It's also an Albanian Qarku/County. That answers what is Gjirokaster.
This is what Gjirokastër is!
Gjirokastër is a really old and small town
Gjirokastër is a 17th-century magical hillside limestone and shale town with spectacular views originally built by the rich landowners of the region.
Gjirokastër is a Qarku!?
It's also one of the 36 former Albanian Districts Rrethi i Gjirokastrës. Now, since the year 2000, it's one of the 12 Counties Qarku i Gjirokastrës.