Translation:She is the third Minister of Finance in this government.
A Minister is a head of a government department. They're usually not elected. They're appointed by the head of government. The person may be a politician, but, more importantly, is supposed to have professional competence in the department they are heading. The Minister of Finance, eg, usually has a background in finance and economics. Just a sidenote regarding US agencies, the Secretaries of US agencies don't sit there and take minutes. They're the heads of the agencies.