The state is the form a human community that (successfully) lays claim to the monopoly of the legitimate use of physical violence within a given territory.--- Max Weber, "Politics as Vocation," (33), translated Rodney Livingstone.
In context, Weber claims that the monopoly is a consequence of a historical process in which intermediary powers and institutions lose their capability for independent use of physical violence. He leaves it a bit ambiguous if it is appropriate to call earlier 'states,' which lacked the monopoly, 'states' or if the use of 'state' has shifted (and that his definition is, thus, context specific). While the idea of a monopoly of physical violence arguably goes back to Bodin or Hobbes, I suspect Weber is the first to use the economic language; this is no coincidence. Throughout the famous lecture, Weber treats the corporation and the state as analogies (in the way that Hobbes uses the family as the relevant analogy for the state). Weber does not address the question why such a political monopoly is a good thing.
Before I get to the implications of this definition, it is useful to notice two absences in it: first, unlike post 1648-Westphalia theorists of international relations, he does not build an explicit recognition of other states into the definition. This is not to deny that other states are relevant here; the monopoly extends to a given territory (and so excludes other powers from it), and so this entails that the boundaries need to be identified not, to be sure, by the international community as such, but certainly by the relevant neighbors who agree on the boundaries of the given territory. (This is so because Weber does not conceive of a world state (in context).)
Second, it is notable that unlike see Hume, Smith, Madison, and Mill, Weber does not explicitly treat the role of (good) opinion (of the authorities, or legitimacy) by the people. He seems to imply that force is the foundation of the state. And, in fact, it is pretty clear throughout the lecture that for Weber politics is the decision of what use of force is legitimate (because lawful). The people with power get to decide what is legitimate (and , Weber is not deluded about this, they will do so generally to benefit themselves).
Yet, it is not as if opinion and language are absent. After all a state also involves a speech act (‘claims’). (This is why declarations of independence and such are so important). And the speech act is, in a certain sense, uncontested. If it were contested there is no claim to monopoly. So, this entails that there is some implied recognition by the subject of the monopoly (that is the state) of the claim. (The recognition may be fairly minimal, but it is not nothing.) The recognition need not involve approval, or a good opinion, but it must involve a de facto acceptance of the monopoly, first, by the rulers who must remain unified, and second, by the rules, who must acquiesce in some not quite trivial sense.
To conclude, then, even this hard-nosed realist approach accepts that the state monopoly on violence does not come in isolation: neighboring states are required for uncontested borders; and there is a common language that presupposes title to the monopoly and the granted authority by those that conform or accede to it.