The Black Council, as it was called, was the Small Council of Rhaenyra Targaryen during the great Targaryen civil war known as the Dance of the Dragons, in which she fought her half-brother Aegon II Targaryen for the throne. They were the leading body of her faction, known as "The Blacks". She formed it in response to Aegon II taking over their father's Small Council in King's Landing itself, which was correspondingly known as the Green Council.
The Green Council was mostly formed of the members of King Viserys I's Small Council at the time of his death - which had come to be dominated by the Greens, the court faction of Viserys I's second wife, Queen Alicent Hightower. Of the seven men on that Small Council, only one was still openly loyal to Rhaenyra: Lyman Beesbury, the old Master of Coin who had served at court for many years and was determined to honor Viserys I's direct orders that Rhaenyra would succeed him to the throne. At the emergency Small Council meeting called upon Viserys I's death in his sleep, Beesbury alone refused to acquiesce to Aegon II usurping the throne - upon which Criston Cole promptly snuck up behind him and slit his throat.
In response, Rhaenyra formed her own Small Council from her seat on Dragonstone island. Due to the death of Beesbury, none of them were previous members of the old Small Council. Indeed it was a much more ad hoc council and it doesn't appear she formally appointed members into all seven traditional Small Council positions.
Rhaenyra's Small Council were her own immediate advisors at court, not necessarily all of the major leaders of "the Blacks" during the civil war, i.e. Cregan Stark wasn't a member of the council but continued to rule at Winterfell while contributing House Stark's armies to the war.
At the outset of the Dance:
- Prince Daemon Targaryen - Rhaenyra's uncle and second husband, younger brother of her father King Viserys I. One of the most feared warriors and military commanders of his generation. Rider of the feared dragon Caraxes.
- Lord Corlys Velaryon - one of Rhaenyra's primary supporters, commander of the large Velaryon fleet which gave Rhaenyra control over the eastern coasts of Westeros.
- Princess Rhaenys Velaryon, born Rhaenys Targaryen, Corlys's wife, granddaughter of King Jaehaerys I, and rider of the dragon Meleys.
- Rhaenyra's three eldest sons, from her first marriage to Corlys and Rhaenys's son Laenor Velaryon:
Rhaenyra's youngest two sons, Aegon the Younger and Viserys II, were only little boys and not included in the council.
Rhaenys, Jacaerys, and Lucerys died during the first half of the war, but then Rhaenyra's faction captured King's Landing from the Greens and held it for about half a year. It is unknown if she more formally established other offices of the traditional Small Council during that time.
In the books
Rhaenyra's rival Small Council was much more loosely formed than Aegon II's (which was largely inherited from the end of their father's reign): it had more than seven members as just "advisors" while not bothering to formally appoint all seven traditional positions.
Daemon Targaryen was named Protector of the Realm in Rhaenyra's stead, meaning he was the formal commander of her armies (usually the title is held by the monarch personally but at times has been handed to a more experienced commander, which Daemon certainly was). Daemon was not Rhaenyra's Hand of the King, however: that office was given to Corlys Velaryon (in this case termed "Hand of the Queen"). Corlys may have also doubled as Rhaenyra's Master of Ships but it hasn't been stated if she formally gave out the position.
The maester appointed to Dragonstone who had served Rhaenyra faithfully for many years was Gerardys. He was also on the council and apparently she considered him her Grand Maester, though in practice a monarch cannot hand out that title, only Conclave can.
While Viserys I's Small Council was not split in the civil war (except for Beesbury, who was killed immediately), the Kingsguard did split between the two factions: four sided with Aegon II while three fled and joined Rhaenyra on Dragonstone. These three become the core of Rhaenyra's rival Queensgard - chief among them was Steffon Darklyn, whom Rhaenyra appointed as its Lord Commander.
There were also about half a dozen other minor lords of the lands around Blackwater Bay who were bannermen to Dragonstone itself that also made up the Black council. They were not significant players in the narrative, and the Season 5 Blu-ray featurette on the Dance of the Dragons simply leaves out mention of them. These included Bertimos Celtigar of Claw Isle, Lord Staunton of Rook's Rest, Lord Darklyn of Duskendale, Gormon Massey of Stonedance, and Lord Bar Emmon of Sharp Point.