"native or inhabitant of Denmark," early 14c. (in plural, Danes), from Danish Daner, (Medieval Latin Dani), which is perhaps ultimately from a source related to Old High German tanar "sand bank," in reference to their homeland, or from Proto-Germanic *den- "low ground," for the same reason.
It replaced Old English Dene (plural), which was used of Northmen generally. Shakespeare has Dansker "a Dane" (c. 1600). Dane was applied by 1774 to a breed of large dogs.
Danegeld (attested from 1086; it was first imposed in 991) supposedly originally was a tax to pay for protection from the Northmen (either to outfit defensive armies or to buy peace), continued under later kings for other purposes. Danelaw (c. 1050) was "the body of Danish law in force over that large part of England under Viking rule after Alfred's treaty in 878;" the application to the land itself is modern (1837, Danelagh).