From A Wiki of Ice and Fire
The Great Council is a council that is formed when the inheritance of the throne is unclear. In Westeros history, there have been at least two Great Councils to decide the Targaryen succession to the Iron Throne.
Great Council in 101AC
Prelude to the Council
King Jaehaerys I's eldest son and first heir Aemon had died in 92AC, which cause Jaehaerys to choose his second born son Baelon over Aemon's daughter Rhaenys as his new heir. Baelon, however, died in 101AC, causing Jaehaerys to be in need for a new heir yet again. Instead of letting the King decide on his own, a Great Council was called together to settle the matter.
The outcome of the Council
Candidates for the crown included Princess Rhaenys Targaryen, Prince Viserys Targaryen, Ser Laenor Velaryon, and Lady Laena Velaryon. The lords assembled favored the male claimant over the female by twenty to one, although supporters of Rhaenys included Houses Baratheon, Stark, Dustin, and Manderly. Viserys Targaryen was ultimately chosen by the assembled nobility.
Following the death of King Viserys I, his small council examined the annals of the Great Council. They determined that the houses which had dissented during the Great Council were likely to support the claim of Rhaenyra Targaryen instead of Aegon II Targaryen.
Great Council in 233AC
Prelude to the Council
King Daeron II's heir, Prince Baelor Breakspear, had perished in an accident during a Trial of Seven in 209AC, leaving his eldest son Valarr to inherit the crown. However, later that year the Great Spring Sickness killed King Daeron and both of Baelor's sons, Valarr and Matarys. This in turn left Daeron's second son Aerys I to inherit. Aerys left no issue, and upon his death in 221AC the crown was passed to his youngest brother, who reigned as King Maekar I, the fourth son of King Daeron II. At the death of Maekar at the hands of a rebellious lord in 233AC, a council was convened to decide who would inherit the Iron Throne since both of Maekar's eldest sons were dead and their inheritors were only children.
Outcome of the Council
Maekar's oldest son, Daeron, called Daeron the Drunken, was a dissolute whore-monger who had died of a sexually transmitted pox, and left only a feeble-witted daughter as a possible heir. His second son, Aerion Brightflame, was a cruel madman who had died in 232AC as a result of drinking wildfire in the mistaken belief it would turn him into a dragon. Aemon and Aegon had not expected to inherit the throne with two grown sons ahead of them. Aemon was already a chained maester serving at his brother Daeron's castle.
The council decided to bypass Daeron's daughter and Aerion's son and first quietly offered the throne to Aemon, who declined the invitation. They then chose Aegon, who was crowned as Aegon V, the fourth son of the fourth son of Daeron II, earning him the name Aegon the Unlikely. Aemon joined the Night's Watch to prevent others from using him against his brother.
Attempts to call a Great Council
In 130AC, after the fall of King's Landing, Dowager Queen Alicent Hightower suggested assembling a Great Council, to decide which claimant should ascend to the throne, Rhaenyra Targaryen or Aegon Targaryen. Rhaenyra, however, refused to grand the whish.
After realizing that Joffrey Baratheon was not the biological son of Robert Baratheon, Catelyn Stark suggests to Renly Baratheon that he, Stannis Baratheon, and Robb Stark, who had all been crowned as kings, should set aside their claims and call for a Great Council to determine who should be king. Renly Baratheon laughed at the suggestion.