Most readers will have some preconceived notion about what the the terms Norse and Old Norse mean. Even if your understanding of the subject is vague, you probably know that Old Norse is famously the language of the so-called Vikings – a historically recent ethnic term that is often used interchangeably with Norse. In casual conversation this works fine, and you're not confusing anybody by using any of the two. Academically (of course) it's much more complicated. If you look in an English dictionary dating to, say, the early 1900's you are likely to see Norse defined simply as «Norwegian». Historically this is what the word meant, but the meaning has since shifted. Today it generally refers to the Germanic speaking Scandinavian population, as well as that of their overseas colonies during the Viking Era and Middle Ages. As for the term «Viking» denoting Norse peoples: This is somewhat misleading because a viking was actually a specific kind of person within Norse culture. A title, really. But owing to its prevalence, scholars frequently resort to it for convenience. For example, Norse overseas colonies in places like Ireland, Greenland, and Russia, is sometimes referred to as The Viking Diaspora. But that's a discussion for a later time. Let's stick with Norse for now.
The first Norsemen
Outside the scholarly world, the term Norse is most commonly applied to the Viking Era, which kicked off in the 8th century for reasons varied and unknown (innovations in sailing technology is commonly assumed to be one factor). The Old Norse language was more or less fully developed by the 700's, having sprung from the preceding Proto-Norse or Proto-Scandinavian tongue (pick whatever term suits you, there's no consensus!), which in turn developed from a Northern dialect of Proto-Germanic.
For Norway, Greenland, the Faeroe Islands, and Iceland, it makes sense to talk about common Old Norse all the way through the 14th century, when the Black Death unleashed a period of Mad Max-esque socioeconomic and demographic turmoil, forcing Norway into a linguistic revolution that gave birth to the Middle Norwegian language. This made mutual intelligibility with Iceland a thing of the past.
Middle Norwegian is basically the sort of grizzly post-apocalyptic linguistic change you can expect when something like three fifths of the entire population shuffles off their mortal coil, leaving their brats with nobody to correct their baby language and youthful slang. Up until then, the differences between Norwegian, Faroese and Icelandic had been a matter of dialect. Specifically the Old West Norwegian dialects of the early settlers.
It wasn't a unified language
Anglophone scholars divide Old Norse language into West Norse, which I described above, and East Norse which encompasses Old Swedish, Danish, and Gutnish (spoken on Gotland). These are distinct enough to be considered languages of their own. Still, all these variants of Old Norse were mutually intelligible to the point where West Norse speakers accepted the term Danish Tongue (dǫnsk tunga) as a name for their own language. This likely originated among Anglo-Saxons to describe the language of Scandinavian settlers, traders and raiders, drawing a generalization from what seemed to them the dominant culture – namely the Danes. This isn't entirely dissimilar to how we generalize by calling them Vikings today. This common linguistic identity, and the fact that they adopted a foreign term as their own, seems to suggest a sense of cultural kinship among Viking Era Scandinavians. Swedes, Danes and Norwegians can still understand each other with relative ease, which is a fantastic linguistic privilege. And still, when reading Norwegian, Danish or Swedish runic inscriptions from the Viking Era, we may enjoy the distinct and recognizable traits of each.
