The Greymarch is an event, or movement, as Sheogorath comes to call it. Occurring at the end of every era in the Shivering Isles, it is intended to destroy the Shivering Isles, Sheogorath's realm. The march begins with large obelisks emerging from the ground, and are the source of power for Jyggalag's forces, the Knights of Order and the Priests of Order.
The lore of the Greymarch comes into place when the other Daedric Lords became fearful of Jyggalag and his immense power. Thus, they decided to battle him, and upon his defeat, cast a curse which would damn him to exist as the thing he hated most: to exist as a Prince who believed in self choice, chaos, and insanity -- Sheogorath. However, the resentful Princes declared that, once an era, Jyggalag could assume his true identity. In this form, he would then destroy the Shivering Isles as a "cleansing" process, and to put order back into the realm.
In the 3rd Era, Year 433, Uriel Septim died, marking the end of the 3rd Era and the beginning of the Oblivion Crisis. Sheogorath, aware of the beginning of the Greymarch, opened a door to his era. An invitation, which the Hero of Kvatch would accept.
The Hero would then conquer the guard of the realm and meet Sheogorath, only to a series of quests revolving on halting Order's plans. After a period of time, Sheogorath had begun the transformation, where he would be reverted into the logical Jyggalag. In his final moments of insanity, he states with disdain that his plan has failed and that he had intended to give the Hero his staff. Thus, Haskill, Sheogorath's chamberlain, will give the Hero a quest to rebuild the staff.
After Sheogorath has transformed back into Jyggalag and he is defeated in one on one combat, he will explain his story and declare the Hero as the new Sheogorath.