Short answers, pulled from the story.
The Danelaw was the part of England where Danish laws applied, covering roughly Yorkshire, the central and eastern Midlands, and the East of England. It arose from the conquest and occupation of eastern and northern England by Danish Vikings in the late ninth century, and lasted in legal form until at least the early 12th century.
The Danelaw originated with the invasion of the Great Heathen Army into England in 865, led by commanders including Halfdan Ragnarsson and Ivar the Boneless. Its legal framework was established between 886 and 890 in the Treaty of Alfred and Guthrum, following Guthrum's defeat at the Battle of Edington in 878.
Alfred the Great was king of Wessex; Guthrum was the Danish warlord who led the final push to conquer it. After defeating Guthrum at Edington in 878 and demanding his baptism as a Christian, Alfred formalized the boundary between their kingdoms in the Treaty of Alfred and Guthrum, permitting Danish self-governance in exchange for loyalty to England.
The Five Boroughs were Leicester, Nottingham, Derby, Stamford, and Lincoln, five fortified towns that broadly delineated the area now called the East Midlands. Four of them became county towns: Leicester for Leicestershire, Lincoln for Lincolnshire, Nottingham for Nottinghamshire, and Derby for Derbyshire.
Norse contact during the Danelaw period contributed the words law, sky, and window to English, along with the third-person pronouns they, them, and their. Contact with Old Norse is also credited with accelerating the reduction of Old English's inflectional system, helping push the language toward the analytic grammar of Middle and Modern English.
Viking rule in the north ended in 954 when King Eric was driven out of Northumbria. As a legal category the Danelaw persisted longer, appearing in the Leges Henrici Primi in the early 12th century, where it was listed as one of three bodies of law alongside those of Wessex and Mercia.