A civil, non-paramilitary police force consisting of police officers called constables. This is the usual definition in Britain, in which all county police forces once bore the title (and some still do).
A military or para-military type force consisting of soldiers trained for police duties. These were mostly established by the United States in the several countries it had protective status over (e.g. Philippine Constabulary, United States Constabulary in West Germany after World War Two, Nicaraguan National Guard, Panama National Guard). These forces also performed military functions by maintaining "mobile forces" of organised units. In Europe such forces are called Gendarmeries or (in Italy) Carabinieri.