The concept of a Nation State has, in the context of modern societies, two particular meanings:
Firstly, the idea of countries ("Nations") with relatively fixed geographic borders. This, in itself, is not necessarily a feature of modern societies, as such, since it is possible to see the development of Nations in pre-modern society, but it is, nevertheless, an important idea when considered in the context of the next point.
Secondly, of more significance here is the idea of "the State" as a political entity characteristic of modern society.
In this respect, the modern Nation State represents a distinct set of social institutions, whereby "the State" has the political authority to create the rules by which a society is governed (which also includes, by-the-by, a monopoly on the legitimate use of violence). Clearly, by "the State" we are referring, in the abstract, to a particular set of social relationships (both institutional and individual), rather than some mysterious non-human entity. In this respect, we can loosely characterise the State in the following way:
Firstly, it involves some notion of government (whether this be democratic or totalitarian), in the sense of political institutions and representatives who have the power to make decisions affecting the whole of society (social policies, laws and the like).
Secondly, in addition to some form of decision-making machinery, the State also involves some way of implementing decision-making and this usually takes the form of a Civil Service (a form of bureaucratic organisation in modern societies).
Thirdly, the State also includes some way of enforcing social controls, through the machinery of a judiciary, an organised police force, armed forces and the like. In addition, writers such as Giddens argue that a feature of modern societies is the way the State employs systematic forms of surveillance in relation to a given population.
Thus, by "the State" we mean a particular form of social organisation and the concept of a "Nation State is significant in relation to modern society in two basic ways:
Firstly, through the concept of nationality (the sense of the individual belonging to a particular national group). This idea is explored further in relation to culture and identity in modern societies.
Secondly, in terms of the distinction between the public and the private spheres.