Bilgoraj
It's a nice little town. Pre-war it had a large Jewish population and Isaac Bashevis Singer set one of his stories there.
I don't know the Polish word for the name of the title "district".
Depends on the size. Lubelskie is a województwo, a large (nowadays) region.
You don't say what country you're from and of course all have different systems which in some cases can vary within the same country.
Województwo is like a large county/region in the UK (say, Norfolk in England or Highlands and Islands in Scotland). I'm not sure if there's an obvious American equivalent to that.
The next tier down is a
powiat, a bit like a district, metropolitan borough or unitary authority in the UK or a county in the US. Bilgoraj, has I think it's own powiat. below that is a
gmina (which the urban part of Bilgoraj certainly has too), like a pre-1974 borough in the UK.
I'm not sure what Americans would call that and of course there are big differences if it's in a conurbation (Warsaw has several powiats and more gminas within the powiats).