In the Anglo-Saxon Chronicle, they're usually referred to as 'The Danes'. When they first show up, they are referred to as 'The Pagans' or occasionally 'Northmen' and talked about in apocalyptic terms, but when the conflict takes on a distinctly political edge with attempts at conquest in the mid-ninth century onwards, 'The Danes' becomes more prevalent. Often, when talking about specific Viking forces, they will simply be referred to as 'The Army' or occasionally 'The Great Army'. Once the Anglo-Scandinavian settlements of the Danelaw are established by the tenth century, the ASC also differentiated between 'The Danes' who come from overseas and the armies from Danelaw sites, usually referencing their county or town of origin. Later specification is also made between the Danes and the Norwegians.

In the Welsh Annales Cambriae, the Vikings are seemingly referred to as Nordmanni or Northmen. It has been previously suggested that this in fact refers to the English of Northumbria, but Dumville points out that the Annales distinguish between the Mercians and West Saxons by calling them Angli and Saxonum respectively, and is likely to have lumped the Northumbrians into one of these ethnic subdivisions rather than politically. It is more likely that the Nordmanni apply either to Vikings or to Anglo-Scandinavian Northumbria.

In Irish annals the vikings are categorized according to their nationality: there are the Danes (Danar), Norwegians (Lochlannaigh) & the "Gall-Gaedil (that is, they are Irish, and fosterchildren of the Norse, and sometimes they are even called Norsemen)", or else they are simply referred to as "foreigners".

Interestingly, in her article The Good, the Bad and the Ugly: Portrayals of Vikings in “The Fragmentary Annals of Ireland”, Clare Downham argues that these categorizations may be later inventions intended to justify the actions of Irish leaders in the viking age (who often allied with viking bands temporarily) by their descendants by "ranking" different viking groups according to their piety and worth. For example, in the Fragmentary Annals of Ireland the Danes are portrayed as near-Christians; in one example they sacrifice to Saint Patrick and Jesus before battle against the Norwegians, although they are still barbarous and semi-pagan. This is important because the Danes are important allies of Cerball mac Dunlainge, the principal "hero" of the text in his adventures against the Norwegians, with whom Cerball later allies with to combat the Gall-Gaedil. This third group is obviously the worst of the three from a medieval Christian audience because they are apostates who have abandoned Christ and "civilized" Irish customs, while the previous two are merely ignorant of Christianity.

This entire scheme of categorization is a complex system meant which appears to serve a narrative rather than an anthropological purpose; it's quite possible that the "Danes" described in the annals several centuries after their adventures could have been a mixture of Norwegians, Danes and other ethnicities (I think /u/Aerandir knows more about this than me), but for the medieval Irish the "ethnicity" was really just a moral rather than ethnic label (though this isn't a universal - the Irish are keenly aware of linguistic and ethnic distinctions between themselves, the Britons & Picts, the Anglo-Saxons and Franks, for example).

This might also be seen in the euphemisms used to describe the Danes and Norwegians throughout Irish texts: Fionnghaill (fair-foreigners) & Dubhghall (dark-foreigners), which might be making reference to moral qualities rather than actual physical features (which many commentators online seem to think). Alternatively, Downham notes that these descriptors may be drawn from genealogical literature, where "fair" was used to describe older lineages while "dark", newer ones. This appellation might suggest that the Irish were aware of groups of vikings who had been in Ireland first (the fair foreigners) who were enemies and rivals of the newer arrivals (the dark foreigners) but made no real ethnic distinction between them, which means that these two categories may be what Irish people during the viking age actually called them.