Adeptus Astra TelepathicaThe "League of Black Ships" is the fleet 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. 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.
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. 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 as well unlike older ones, they can become Space Marine Librarians or Grey Knights.
Navigators are not collected on Black Ships, because the Navigator Gene is familial and the Navigator Houses keep all those genes to themselves on or near Terra. And because the Navigator Houses would go ballistic if a Black Ship did collect their kin. Literally ballistic, as in, Nova Cannon shells of the Navis Nobilite firing on the Black Fleet, the Solar Fleet (the Imperial Navy Fleet of the Solar Segmentum), holding the Solar system (except Mars, which likely would not take one side or another in this case) ransom until the Inquisitors responsible are executed. If this happened, which would not be a Chaos Heresy scenario, the civil war would use up all of the Segmentum's resources which means everybody from Orks, to Tau, to Abaddon (who has been de-failbaddoned by the fluff, that is, someone gave him his arms back) could invade and remove the heart of the Imperium. The Adeptus Astra Telepathica does not want to be responsible for causing the Imperium to lose to the leader of the Black Legion. (not to mention causing the Astronimcon to blink out out because the golden throne got destroyed. (Though, that wouldn't be too bad. It'd stop the Tyranids from coming. But it's also make another Eye of Terror, so win-lose.)
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.