“The state is that human community which, within a certain territory, claims for itself (successfully) the monopoly of legitimate physical violence.”Max Weber · Politica, o vocație · 1919 · paragraful 4
The state is defined by its exclusive monopoly on legitimate physical violence within a territory.
Weber offers a sociological definition of the state, not a legal or moral one. What makes the state unique is that it alone has the right to use force (police, army) and to authorize others to do so. This monopoly is 'legitimate' because society accepts it as such. For example, a citizen cannot take justice into their own hands, but the state can arrest and punish.
Why it matters This definition is essential for understanding why the state has authority, but also why abuse of power or private violence (mafia, terrorism) undermines the very essence of the state.