A nation-state is a political unit where the state, a centralized political organization ruling over a population within a territory, and the nation, a community based on a common identity, are congruent. It is a more precise concept than "country", since a country does not need to have a predominant national or ethnic group.
Portrait of "The Ratification of the Treaty of Münster", one of the treaties leading to the Peace of Westphalia, where the concept of the "nation state" was born
A state is a political entity that regulates society and the population within a territory. Government is considered to form the fundamental apparatus of contemporary states.
Painting of Roman Senators encircling Julius Caesar
The frontispiece of Thomas Hobbes' Leviathan