Adeptus Astra TelepathicaThe "League of Black Ships" is the second-largest armed fleet in the galaxy, after the Imperial Navy. They fall under the purview of the Adeptus Astra Telepathica, and its function is to gather psykers for processing on Terra. They stop at a planet and tell the governor to hand over all the psykers he has as part of the Imperial Tithe. OR ELSE.
Because of the extreme volatility of their cargo, every cell is heavily shielded and warded; even then, sanity-blasting horror awaits those who enter, and the fates of those gathered is almost inevitably painful death of one form or another. Crews of Black Ships are stringently tested and monitored for any possible spark of psychic potential and are routinely mind-scrubbed to limit any corrupting exposure from the tainted cargo that the ships carry.
It transpires that a significant portion of the defunct Sisters of Silence found employ in the Black Fleet after their dissolution following the Horus Heresy, considering it was always part of their function prior to the civil war, it makes sense that enough of them would remain to continue the role, even after the organisation was disbanded. The fact that they never received a mention up until the return of Roboute Guilliman could be attributed to the level of secrecy afforded to the Black Fleets in general; virtually no-one outside of the Adeptus Astra Telepathica knows how large the fleet is, or the route of planets that they take. Additionally, very few outside individuals are ever accommodated passage on a Black Ship, only the Inquisition or other high ranking individuals such as Space Marines or Sisters of Battle may ever be allowed to board, and only at the discretion of the captain.
Their journeys can be long, but there are so many in service, and so many being brought into service all the time, that there is a constant stream of thousands of psykers arriving at Terra every day.
While they are only ever a background organization in the fluff, their importance to the Imperium is paramount, not least due to the fact that they harvest potentially dangerous and untrained psykers from across the galaxy. But if the flow of psykers to Terra was to ever stop or slow down, the Astronomican would eventually fade out due to its constant need for replenishment. Additionally, the Golden Throne itself would cease functioning, as its own separate need for psychic sacrifices need to be met. In fact, since M36 the number of psykers required has increased four-fold since the Emperor's entombment. Combine that with the Great Rift splitting the Imperium in half, thus leaving only half an Imperium's worth of psykers to reliably harvest and transport back to Terra, and the Black Fleet is more busy now than ever before to keep the Astronomican supplied.
Not all psykers "acquired" in this manner are sacrificed, mind you. Those who don't go insane on the ride to Terra, or those who are exceptionally powerful, are often sanctioned and sent to fight amongst the Imperial Guard or alongside the Inquisition in one of its many arms. Those that are not the very cream of the crop but still good enough to serve end up as Astropaths. Many of these psykers are collected as children, too, because if they are powerful or stable enough, they will be more impressionable and easier to indoctrinate and unlike older ones, the strongest of youngsters can be handed over to a Space Marine Chapter to undergo the trials to become a Librarian (Or for the really exceptional ones, part of the Grey Knights).
Navigators are not collected on Black Ships: Despite being psykers of a sort, their lineage is carefully cultivated by the Navigator Houses, which are tightly bound to the service of the various institutions of the Imperium. Since their genes are recessive and vanish after one generation of interbreeding with baseline humans means that Navigators simply do not arise on worlds where they do not already hold sway, and never as random mutations. Rogue Navigators do happen, both as individuals and as entire houses running renegade, but in those cases they are just as likely to get *BLAM* for forsaking their mandated duties. This all means that the worlds where the Black Ships visit are not likely to even have any Navigators to hand over for processing.
Inquisitors usually get around by seizing other peoples' ships, but the Inquisition does maintain a fleet of its own special ships. These Black Ships are rather like the archetypal Black Helicopters of conspiracy theory fame: though rarer than their military cousins, they are specially-built to be faster, quieter, and deadlier, and employed for sinister purposes.
In battle, most of these ships have their running lights turned off and go to minimal power emissions, contrary to Imperial Navy warships which turn on all their lights and have their weapon systems clearly swiveling and powering up to be more intimidating. Though few have actually witnessed a Black Ship in battle, conflicting reports say they are usually of a class of vessel mimicking the oldest designs (horizontally oriented instead of vertical, and without the distinctive armored prow of 41st millennium Imperial Navy ships, basically the kind Chaos fleets use most of the times), and painted as black as the space they sail through. They state that at one moment a Black Ship will appear to be completely defenseless, then quick as blink, suddenly bristle with weapon and lance batteries, as well as types of weapons not conventionally employed by Navy ships.
Most of the time they are parked over a planet where an Inquisitor is performing a high-profile and important investigation, both to intimidate the local population and to perform Exterminatus when push comes to shove. Inquisitors tend to tag along on the other sort of Black Ships as well, even though they are legally under the control of the Adeptus Astra Telepathica. This lets them keep an eye on the tremendously-dangerous cargo and inspect any worlds tithed along the way.
Other uses of the term
Black Ship was a term used that the Japanese to refer to western ships between the 16th to 19th centuries, in particular it was used to describe steamships which began to appear in the 19th century, including those of Commodore Perry's fleet in 1854. The Black Fleet is the assumed name of the ancient Greek warfleet from the legendary Trojan War in Homer's recounting of it, the Illiad.