Modern state refers to the idea of linking the notion of "political" to the exercise of (independent) sovereign authority. It is necessary to understand it's distinction from the notion of premodern state. In a modern state the political and legal authority is with the ( impersonal) state unlike the premodern where it is not clear who actually is the holder: may be king /church, clergy/society...
It did not come in a day or on a a certain date. it is said to have evolved but two landmark events laid the foundation specifying features of a modern state namely Treaty of Westphalia 1648 and Montevideo Convention 1933. Study of social contract theory and classical liberal thinkers philosophy to newer version of state would help in this regard.