Sweden is Europe's 5th largest country, or about the size of California. For population data, as well as general up-to-date information about Sweden and its history, please visit Wikipedia.
The map to the right shows the province boundaries in Sweden. Provinces date back to the Middle Ages, when they each had their own rulers and laws, and were eventually united into the country of Sweden. In the 17th century they were in some ways superseded by counties, that have been used for administrative purposes since then (e.g. in organizing health care today).
Provinces still exist today, though for cultural purposes. One official use is in royal titles; the Swedish equivalent of e.g. "the Duke of Sussex" refers to a province, not a county. But above all, it's very common to relate one's cultural identity to a province rather than a county. Cultural heritage and tourist information also tends to refer to provinces.
Provinces and counties often share borders, but genealogists typically prefer to use provinces, as counties have been less stable over time (splitting or merging according to political whims—the latest batch of changes was in the 1990s).
P. Alfred Tranberg Klingspor's parents lived all their lives in the southmost part of Sweden. His father, artilleryman Nils Carlsson Tranberg, came from Skåne, and his wife Petronella Andersdotter from neighboring province of Småland. They first lived in Skåne, but moved to Blekinge while Alfred was a baby.
On Vendela's side, all of her ancestors in the last four generations had lived in Östergötland (right north of Småland).
During their marriage, P. Alfred and Vendela lived in various towns in Östergötland for the first four years. Then they moved to Gothenburg (where Västergötland reaches the west coast), where they separated a couple of years later. Vendela and Laura then lived in Östergötland for seven years before they emigrated.
P. Alfred seems to have lived in Gothenburg for the rest of his life, except for the 1890s, when he lived in Skåne and met his new partner.