Län and lääni are the Swedish and Finnish language terms, respectively, for the administrative divisions used in Sweden and Finland. They are also sometimes used in other countries, especially as a translation of the Russian word oblast. During the period when Finland was a part of the Russian Empire (1809-1917), when Russian was made an official language along side of Swedish it was synonymous with the word guberniya.
The word literally means fief. The usual English language terms used are separate for the two countries, where Sweden has chosen to translate the term to "county" while Finland has chosen "province".With a shared administrative tradition spanning centuries, ending only in 1809, this is a separation by convention, rather than by distinction. The term matches reasonably well the British term "county", but not so well the American term "county" which are much smaller in size, more like Swedish "kommun".
The reason why Sweden has chosen to translate the term to "county" is that in Swedish and English, the word "provins"/"province" has come to mean different things. In the Swedish Empire, all lands conquered became provinser (provinces); Swedish law, which granted the common people much more freedom and influence than any other European law at the time, was not extended to them, remaining confined to the landskap (in plural) which made up the Swedish-and-Finnish heartland (roughly corresponding to present-day Sweden and Finland). Examples of such former Swedish provinser are Estonia and Swedish Pomerania. Another reason is that in education, Sweden has preferred British English over American English. "County" is a reasonable British English translation of Län.
In both countries a län/lääni is but an arm of the executive power of the national government, and has no autonomy nor legistlative power. The län/lääni subdivision does not always match the traditional provinces, which are called landskap (singular and plural) in Swedish (including Swedish-speaking Finland) and maakunnat (singular maakunta) in Finnish.
The governor has the title landshövding (Swedish), maaherra (Finnish). He or she is appointed by the government, and presides over the länsstyrelse (Swedish) or lääninhallitus (Finnish) - translated as "County Administrative Board" and "State Provincial Office", respectively. The governor's office is administrative by nature, which is also hinted at by the now obsolete title Konungens befallningshavande - "the King's Deputy" - and traditionally used as an honourable post for politicians to conclude their careers.