Depending on the context, a foreign intelligence service is either an intelligence service that specialises in foreign intelligence (as opposed to security intelligence), or an intelligence service that belongs to a foreign country. The latter shouldn't require any more explaining, however I'll go into more detail about the former: Foreign Intelligence is intelligence that relates to foreign entities, whereas security intelligence pertains to domestic security issues. However, this should not be taken to mean foreign intelligence = overseas, security intelligence = domestic. Both kinds of services generally do both overseas and domestic operations, although usually for different purposes. This inevitably leads to a small degree of crossover (such as Counterterrorism or Counterintelligence), which often results in rivalry between agencies (MI5 vs MI6, for example).
Some notable Foreign Intelligence services:
*Secret Intelligence Service (SIS, better known as MI6 - Military Intelligence, Section 6) | Britain
*Australian Secret Intelligence Service (ASIS) | Australia
*Central Intelligence Agency (CIA) | USA
*Canadian Security Intelligence Service (CSIS) | Canada
*Foreign Intelligence Service (SVR) | Russia
Some notable Security Intelligence services:
*Her Majesty's Security Service (Better know as MI5 - Military Intelligence, Section 5) | Britain
*Australian Security Intelligence Organisation (ASIO) | Australia
*Canadian Security Intelligence Service (CSIS, deals in both security and foreign intelligence) | Canada
*Federal Bureau of Investigation National Security Division (FBI NSD) | USA
*Federal Security Service (FSB) | Russia