A cardinal was a senior official of one of the six Gallifreyan chapters. (PROSE: Doctor Who and the Deadly Assassin) Each had multiple cardinals, who met as the Chapter Council. (PROSE: Lungbarrow) A number of cardinals, but not all, sat on the High Council. (AUDIO: Spring, Lies)
The most senior cardinal was known as the Cardinal Prime. (PROSE: Lungbarrow) Somewhere between Under Cardinal (A sub office of Cardinal, ranking below it) and Cardinal, there was the rank of Junior Cardinal, a rank held by Borusa at the time of the Morbius crisis. They wore grey robes. (PROSE: Warmonger)
The First Doctor's family hoped that he would one day attain the rank of cardinal, making him the first of his House to do so. However, he had other plans for his future. (PROSE: Lungbarrow) His brother, Braxiatel, later became a cardinal. (AUDIO: Weapon of Choice)
Chancellor Borusa told the Fourth Doctor that he would see what he could do to persuade the cardinals to accept him as President as politeness dictated. The Doctor pointed out that there were no other candidates and the cardinals need not be consulted. The Gold Usher welcomed them to the Doctor's investiture. (TV: The Invasion of Time)
By the time that the Time Lord Academy was opened to outsiders, the High Council was, according to Cardinal Braxiatel, made up of three cardinals who were unsympathetic towards President Romana. (AUDIO: Lies)
Cardinal Borusa was the leader of the Prydonian Chapter when the President was assassinated by Goth. (TV: The Deadly Assassin) He was later made Chancellor by the High Council. (TV: The Invasion of Time)
Both Ollistra (AUDIO: Only the Monstrous, The Side of the Angels) and Padrac (AUDIO: The Eleven) were Cardinals during the Doom Coalition crisis. Ollistra continued as Cardinal through into the Time War.
Franilla, Sendok, Zass, (PROSE: Divided Loyalties) Helron, Pandad, (COMIC: The Legacy of Gallifrey) Leabind, Tachen, (AUDIO: Lies) Luvis, (AUDIO: Omega) Morissa, (AUDIO: Lies) Tortheth (PROSE The Quantum Archangel) and Undercardinal (AUDIO: Neverland) were all Cardinals at one point.
One Cardinal wrestled with the Garvond in the Matrix. It ate away his remaining lives, killing him without hope of regeneration. This was recorded in The Worshipful and Ancient Law of Gallifrey. (PROSE: The Dimension Riders)