"Dane" could mean different things depending on who was using the term and how. In most of our Western European sources (Latin and Anglo-Saxon), it simply refers to anyone who spoke Old Norse ("Danish") regardless of whether they were visiting Scandinavians, immigrants from Scandinavia, indigenous people or even roving vagabonds who acculturated to one of the spreading "Danish" cultures, or the descendants of any of such groups. Old Norse quickly became the common tongue of the North Sea and the North Atlantic Archipelago during the 800s, so "Dane" rapidly became a very vague term indeed. At the same time, however, kingdoms were coalescing in different parts of Scandinavia, including one straddling the Danish archipelago and extending into southern Sweden and northern Germany. Since this was the nearest kingdom where "Danish" was spoken, it appears in Frankish sources as the "kingdom of the Danes"—despite the fact that this was actually a gross simplification of what must have been a very complex political situation indeed.

So who actually made up those viking war bands that still grab our attention 1200 years later? To borrow from an earlier response, most of the textual accounts suggest that viking bands were raised by chieftains recruiting from their home fjord. Archaeological evidence contradicts this. So far, we've been able to do the necessary isotopic analysis of four viking cemeteries—9th-century Dublin (n=4), 9c Repton, England (n=3), 10c Ridgeway, England (n=38), and 10c Trelleborg, Denmark (n=52). Although this is only a small statistical smattering from the Viking Age, the results are unanimous: whether at home or abroad, viking warbands had diverse origins. Of the 52 individuals buried at Ridgeway, at least 38 came from points outside the UK, including Iceland, the Baltic states, and Belarus. It's hard to imagine what these diverse groups would have spoken, and one recent study suggests that each group developed its own unique pidgin version of Old Norse as a lingua franca. War leaders counteracted this heterogeneity by actively building group cohesion. The authors in the article above point to material markers (e.g. carrying similar weapons), oath-taking, and shared ideology (e.g. adopting a Norse paganism).

For a very good overview, check out D.M. Hadley, The Vikings in England: Settlement, Society and Culture (Manchester: Manchester University Press, 2006). I'd also recommend the Oxford Illustrated History of the Vikings or Vikings: The North Atlantic Saga as two very accessible and richly illustrated introductions.