Falun is today a small town, in the province of Dalarna, county of Kopparberg (formerly, it's renamed Dalarna's county these days).
Today the municipality has 54000 inhabitants, of which 36000 live in the town proper.
Falun was granted town privileges in 1641, but had been an important place (due to being the copper mining capital) since the late Middle Ages. Around the mid 17th century it had 6000 inhabitants and was thus one of the most populous towns in Sweden.
Sweden - which is to say Falun - was the biggest producer of copper in Europe at the time. So big and important was the copper mining that Sweden's economy was not based on the gold standard but on the copper standard!
The mine at Falun has been declared a World Heritage Site.
You can read more about Falun (in English of course) at
http://www.falun.se/www/english.nsf Today Falun municipality has ten parishes but I think that maybe only one or two were designated Falun in the old days.
The ten parishes are: Aspeboda, Bjursås, Enviken, Falu Kristine, Hosjö, Stora Kopparberg, Sundborn, Svartnäs, Svärdsjö and Vika.
The ones I suspect of being Falun of old are Falu Kristine and also Stora Kopparberg.
Ingela