It was originally created in 1E 221 after the exceptionally long reign of King Harald, who died at 108 years of age and outlived all but three of his sons. Over the years, however, the Moot became permanent and acquired an increasing amount of power. By the reign of King Borgas, the last of the Ysgramor Dynasty, the Moot had become partisan and ineffective.
After the murder of King Borgas by the Wild Hunt, the Moot's failure to appoint Jarl Hanse of Winterhold sparked the disastrous War of Succession, during which Skyrim lost control of its territories in High Rock, Morrowind, and Cyrodiil, never to regain them.
The Moot has also been used in a ceremonial role. When High King Istlod of Solitude died, a moot was convened to name his son and legitimate heir Torygg as the next High King. This took place despite the fact there being no dispute as to who should ascend to the throne.
This type of Moot is more of a formality and theater than an actual meeting to choose a king, only because in recent times the High Kings have sworn fealty to the Emperor. Since Solitude is the city most directly influenced by Imperial culture and politics, the Jarl of Solitude has served as High King for generations.
In the intervening millennium, the Moot had only been called upon three times when the throne was in dispute, and succession has never been decided on the field of battle - even when Skyrim became part of the Empire.
However, in 4E 201, High King Torygg was killed by Ulfric Stormcloak in a challenge to the throne. This event plunged Skyrim into a civil war, and for the first time in centuries, a fourth moot will be called to elect a successor. The victor of this civil war will likely determine the next High King of Skyrim. If the Dragonborn aids the Stormcloaks to victory, the choice will probably be Ulfric Stormcloak. If the Dragonborn decides to assist the Empire instead, then Elisif the Fair, Torygg's widow, will be the top candidate.
As of the Dragonborn expansion to the game, the actual moot and election of High King has not occurred in the game's timeline and likely will not happen.