The Danelaw was an area that covered the north and east of England during the 9th and 10th centuries. It was mainly controlled by Danish (not Norwegian) Vikings, and the Dane's law was used to rule the people (hence Danelaw).
The Danelaw area came about because the Anglo-Saxon kingdoms were not powerful enough to beat back the arrival of the Danish army under King Guthrum. The only real resistance was by the Saxons, and even they were beaten back at first. The only reason they did not overrun the west was because the Danes were supplied from the east, and the west of England was simply too far for them to get supplies to their army.
This gave the Saxon King Alfred the time to regroup and form an effective defensive army.
Eventually, the strengthened Saxon army fought an important battle against the Danes which resulted in a kind of truce, with the Danes continuing to hold the Danelaw area, but never again attacking the west and south.
By the 11th century, the whole country had merged into one kingdom, and the Danelaw went out of use.