Kingsmoot

The Kingsmoot is an ironborn custom and part of the traditions which they call the Old Way. Unlike regular lordships or the kings of the mainland, in past centuries the King of the Iron Islands was not a hereditary monarchy, but an elective kingship: all the lords and ship captains of the Iron Islands would gather at an assembly known as a Kingsmoot to elect a new king from among their own numbers.

History
After King Harren the Black and his sons were killed in the War of Conquest, Aegon the Conqueror had the ironborn select a new House to rule the Iron Islands (under the overall rule of the Targaryens), according to their custom. The ironborn chose Vickon Greyjoy as Lord of the Iron Islands, and House Greyjoy continued to rule over the ironborn by hereditary right for the next three centuries. Vickon Greyjoy's election, however, was not called a "kingsmoot". There has not been a formal kingsmoot in thousands of years.

In the books
In the A Song of Ice and Fire novels, during a Kingsmoot all captains may cast a vote to choose the new Iron King and any captain may put his name as candidate. Kingsmoots are traditionally held on Nagga's hill on Old Wyk. According to the book History of the Ironborn there hasn't been a kingsmoot in four thousand years, but according to Questions, it has been two thousand years. Some famous Iron Kings were chosen at a Kingsmoot: Urras Ironfoot, Sylas Flatnose, Harrag Hoare (of House Hoare), and the Old Kraken (of House Greyjoy).

Once a candidate puts himself forth, he is bound to the kingsmoot decision. Only once in the history of the Iron Islands has the election been declared unlawful. When one king died, the king's brother called a kingsmoot while Torgon the Latecomer, the deceased king's son, was off at sea raiding. The king's brother hoped he would be elected in the son's absence, but the kingsmoot raised up Urragon Goodbrother, whose first act was to put all the present kin of the old king to death.

Torgon returned two years later and claimed the kingsmoot was unlawful since he, the previous king's son, was not present to put forth his candidacy as was his right. Since Urragon was bloodthirsty and his cruelty made him many enemies, the priests and Ironborn denounced him and raised Torgon to be the new Iron King.

The last Kingmoot was a bloodbath and took place centuries before the Andal invasion. Urron Greyiron gathered his men and killed all the other attendees and set up the Greyiron dynasty, which lasted a thousand years until the arrival of the Andals.

Following Balon Greyjoy's death, his brother Euron attempts to take over the Iron Isles. Aeron Greyjoy, who loathes Euron for being a godless man (and perhaps for more personal reasons which are implied in his nightmares), forestalls him by announcing that Kingmoot must be held. The ironborn, who respect Aeron greatly, obey him. Aeron was also personally opposed to the idea of Balon's daughter Asha (called Yara in the TV series) directly inheriting rule from her father, because he felt it was ungodly for the ironborn to be ruled by a woman. Nonetheless he respects the tradition of the Kingsmoot enough that he will not outright deny Asha the right to put forward her candidacy, he just hopes she won't win.

In Season 6's "Home", Aeron implies that the succession law of the revived kingdom of the Iron Islands is clear that a Kingsmoot must be held. Actually, the kingship had become hereditary centuries before the Targaryen Conquest, and upon Balon's death there is an open question of who should rule - Asha/Yara or one of her uncles, because the inheritance laws for their new kingdom hadn't been clearly established (though in private Balon indeed wanted Asha to rule after him). Of course, there isn't necessarily a discrepancy: just because TV-Aeron says "the law is clear" doesn't mean it actually is, this could be his opinion that he is forcing on everyone else as a powerful clergyman. That is, in the novels, Aeron similarly says that they "must" have a Kingsmoot and the ancient traditions are clear on this, even though they actually aren't - he really wants to hold a Kingsmoot in the hope that Asha might not win it.