Short answers, pulled from the story.
Conciliarism was a movement in the 14th, 15th, and 16th century Catholic Church holding that supreme authority rested with an ecumenical council rather than the pope. It held that a council could act apart from, despite, or even against the pope's wishes.
Conciliarism emerged in response to the Western Schism, when rival popes reigned at once in Rome and in Avignon. Proponents argued that both popes should abdicate to allow a single new election.
The Council of Pisa met in 1409 but failed to end the schism. The Council of Constance met from 1414 to 1418 and had more success, and Conciliarism reached its peak at the Council of Basel, held from 1431 to 1449.
The Fifth Lateran Council condemned Conciliarism between 1512 and 1517, confirming the pope's power and teaching authority. Pope Pius II had already opposed the movement through his bull Execrabilis, and Pope Pius VII condemned the conciliarist writings of Germanos Adam on the 3rd of June 1816.
William of Ockham and Marsilius of Padua laid early theological groundwork, with Marsilius arguing in his 1324 work Defensor Pacis that the universal Church was made up of the faithful, not the priesthood. Later theorists Jean Gerson, Pierre d'Ailly, Francesco Zabarella, and Nicholas of Cusa developed a more clerical version of the theory.
Conciliarist strains of thought remained within the Catholic Church, and interest revived with the convocation of the Second Vatican Council in the 20th century. Professor David D'Avray noted that the council's documents emphasize episcopal authority conjoined with papal authority rather than above it.