Black sites were clandestine prisons or installations operated by powers such as the Romulan Star Empire, the Klingon Empire, and the Cardassian Union either presently or in the past. These prisoners were used to house prisoners or materials they considered too dangerous to be publicly acknowledged.
In the United Federation of Planets the use of these types of clandestine prisons were explicitly forbidden under the Articles of Federation.
During his time with Starfleet Intelligence Tuvok became aware of other Alpha and Beta Quadrant powers using such black sites.
By 2385 the Klingon General Shaniq and those loyal to him had set up a clandestine prison on Nydak II, in opposition to the Klingon government under Chancellor Martok. Federation Pro Tem's chief of staff Galif jav Velk arranged with Shaniq to have Cardassian True Way prisoners detained and tortured for information at that prison.
The black site at Nydak II was destroyed by Klingons loyal to the Chancellor, who considered that black site to be an affront to their honor. (TTN - The Fall novel: The Poisoned Chalice)