Short answers, pulled from the story.
The Ninety-five Theses, also called the Disputation on the Power and Efficacy of Indulgences, were a list of propositions written in 1517 by Martin Luther for an academic debate. They aired theological misgivings about the theory and practice of indulgences and are retrospectively considered to have launched the Protestant Reformation.
Martin Luther wrote the Ninety-five Theses against the church's practice of selling indulgences. He became especially concerned in 1517 when his parishioners returned from buying Johann Tetzel's indulgences claiming they no longer needed to repent or change their lives to be forgiven of sin.
On the 31st of October 1517, Luther sent the Ninety-five Theses with a letter to Albert of Brandenburg, the Archbishop of Mainz, under whose authority the indulgences were being sold. That date is commemorated as the beginning of the Reformation.
The posting is uncertain. Philip Melanchthon first mentioned that Luther posted the Theses on the door of All Saints' Church on the 31st of October 1517, but this conflicts with Luther's own statements that he raised his objections through proper channels, and he may not have posted them on the door at all.
The Theses argued that the pope had no power over souls in purgatory and could release people only from punishments the church itself imposed, not the guilt of sin. Luther taught that true repentance was better than buying an indulgence and that the truly repentant Christian had already received the only benefit an indulgence could provide.
Luther was summoned to Rome, debated Thomas Cajetan at Augsburg in October 1518, and refused to recant. He was finally excommunicated in 1521 after he burned the papal bull threatening him with excommunication, and the controversy became the beginning of the Reformation.