A Black Ship is an FTL-capable starship operating without being placed on the Naval Registry. More specifically, the term refers particularly to military ships being operated in secret by government organizations and by the Gudersnipe Foundation, as these ships are in violation of several treaties.
During the short story Forbidden, the Saratoga encounters a Runarin heavy battleship which does not match anything on record but appears to be of a standard design, indicating that the Runarins may in fact be operating an entire fleet of black ships. This is considered particularly agrevious, as the Runarins are chiefly responsible for maintaining the international naval registry.
Despite it's rather immense capabilities, the Foundation must go to extreme lengths to obtain black ships of it's own. Facilities like Utopia Gregaria, while inside Foundation-space and under absolute Foundation control, is home to some six hundred thousand Alliance citizens, with the ship-building facilities subject to inspection. These inspections are, ostensibly, for health and safety(following the Nezbit Inquiry), but still give the Alliance insight into the goings on of the facility.
Most black ships fielded begin life as failed prototypes, built in-house at Gudersnipe School and fielded only occasionally. For the most part, the Foundation uses civilian ships for it's "black" operations, or they use various strategies to mask the deployment of their registered ships(By treaty, the Foundation is required to report to the registry every time a ship is moved by GATE).