Burning of King Olaf was a festival performed by the Bards long ago in Skyrim's history. The festival included the burning of an effigy meant to represent King Olaf One-Eye, who was sentenced to be burned for betraying Solitude. After the return of the Dragons, around 4E 201, Elisif the Fair, the Jarl of Solitude, prohibited the practice of the ritualistic burning. She felt the celebration was too gruesome, given the death of her husband, High King Torygg and the Civil War conflict between the Stormcloaks and the Imperial Legion.
If the Dragonborn becomes a member of the Bards College, Viarmo, the headmaster, will send them to find the last remaining King Olaf's Verse, a heretical poem written by Svaknir, hoping to prove to the Jarl that the College has a legitimate claim to holding the festival.
The quest begins inside the Bards College with Viarmo, the same target as the quest to return the book from the bard in Windhelm. Upon speaking to him, the Dragonborn is given a dialog option to ask for admission. He then tells the story of King Olaf's Verse, which was thought to be lost when the king ordered the copies destroyed. However, one copy is suspected to be with the bard at Dead Man's Respite. Viarmo then asks the Dragonborn to retrieve the book from the dungeon. The marker for the dungeon shows up on the map, south of Solitude.
The Burning of King Olaf One-Eye is a direct parallel to Guy Fawkes Night, a similar event in the UK, where a 'traitor of the people'--Guy Fawkes-- is burned in effigy annually for their perceived betrayal.