Short answers, pulled from the story.
William the Conqueror (c. 1028-1087) was the first Norman king of England and Duke of Normandy, who defeated Harold Godwinson at the Battle of Hastings on the 14th of October 1066 and was crowned king on Christmas Day 1066. His conquest transformed England's church, aristocracy, language, and ties to continental Europe in ways that persisted through the Middle Ages and beyond.
The Battle of Hastings began at around 9 in the morning on the 14th of October 1066 on Senlac Hill, about six miles from Hastings, and lasted all day. Harold's shield wall initially repelled William's forces, but staged Norman retreats drew English soldiers out of position, exposing them to cavalry attacks; Harold was killed during the fighting and the English defence collapsed.
William was called the Bastard because he was the illegitimate son of Duke Robert I of Normandy and Herleva, who was not Robert's wife. His mother later married Herluin de Conteville, with whom she had two more sons, Odo of Bayeux and Count Robert of Mortain. Despite the stigma, William was recognised as his father's heir and was made to swear fealty by Norman magnates before Robert left on pilgrimage in 1035.
The Domesday Book was a comprehensive survey of landholdings across English counties south of the rivers Tees and Ribble, ordered by William at Christmas 1085 and mostly completed by the 1st of August 1086. It recorded who held each piece of land before and after the Conquest, along with its value, tax assessment, and resources. William's motivations likely included documenting feudal obligations and justifying increased taxation.
William died on the 9th of September 1087 at the priory of Saint Gervase at Rouen, after being injured or falling ill during a campaign in the French Vexin in July 1087. He left Normandy to his eldest son Robert and arranged for his second surviving son William to become king of England, sending him ahead with a letter to Lanfranc the Archbishop of Canterbury.
The Harrying of the North was a campaign William conducted in late 1069 and early 1070 after a combined Danish and English rebel force captured York and Edgar the Aetheling was proclaimed king by his supporters. William marched through the region burning and destroying the countryside as he went, wearing his crown symbolically in the ruins of York on Christmas Day 1069. The campaign ended by April 1070 when William wore his crown ceremonially at Winchester for Easter.