Län and lääni are the Swedish and Finnish terms for the administrative divisions used in Sweden and Finland, and sometimes in other countries, especially as a translation of the Russian word oblast'.
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 a separation by convention, rather than by distinction.
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 limits of the traditional provinces, which are called landskap (singular as well as plural) in Swedish and maakunnat (singular maakunta) in Finnish.
The governor has the title landshövding (Swedish), maaherra (Finnish) and he or she is the head of a länsstyrelse (Swedish), lääninhallitus (Finnish), which translates into either a County Administrative Board or a State Provincial Office depending on the country. The governor is an administrative position, and it's a common post for politicians to "retire" into from national politics.
See also: Letter Ä