So there's a few things to unpack in here.

Who were the vikings? Quite simply the vikings were the Norsemen who took to the seas as pirates and raiders, there were of course Norsemen who went abroad as traders and merchants though the distinction between the two was not set in stone. The word viking is used primarily to describe the pirates in the context of Anglo-Saxon England, as the OE word wicing. There is an inherent martial connotation to viking as opposed to Norsemen however.

The Norsemen were the inhabitants of what we today call Scandinavia, Norway, Sweden, Denmark, Iceland, and other small islands in the North Atlantic such as the Faeroe Islands. Viking raids were first noted starting in the late 8th century with the destruction of the monastery at Lindisfarne in Northern England. The did not suddenly spring into existence at this point of course, pre-viking era Norsemen are well documented archaeologically (indeed many of our modern depictions of Norse helmets for example come from the Vendel period).

They did not exist as "sea nomads". The traders, pirates, merchants, and such that we call vikings made their living on the sea but they did not inhabit the sea as their home. The Norsemen had polities across the areas that the raided, traded with, and inhabited. Norse settlements were located not only in Scandinavia, but also Scotland, England, Ireland, Russia, France, and even in the New World. Their raids and trading would be done in the summer months and for the rest of the year the Norsemen would winter or live in these settlements. Their raids reached as far as the Mediterranean, their trade network stretched from Canada to the the Abbasid Caliphate in Iraq, Norsemen served as mercenaries in the Byzantine Empire. Many more Norsemen likely never took to the seas as warriors or merchants but lived out their life as farmers.

However they did not live in "nations" as we understand them. Early on in the viking period, Scandinavia was extremely decentralzied with most of the populace probably being dominated by a small warrior elite but with extremely limited geographical reach. Over the course of the viking age though these small polities gradually coalesced into larger realms such as the kingdoms of Norway, Denmark, Sweden, and the commonwealth of Iceland. Other viking ruled areas such as Normandy, the Danelaw in England, and the Rus polities in Russia gradually assimilated into their surrounding areas. Though it does bear mentioning that at the height of Scandinavian power one monarch, Cnut the Great, controlled almost all of the North Sea.