A failed state is a political body that has disintegrated to a point where basic conditions and responsibilities of a sovereign government no longer function properly (see also fragile state and state collapse). A state can also fail if the government loses its legitimacy even if it is performing its functions properly. For a stable state, it is necessary for the government to enjoy both effectiveness and legitimacy. Likewise, when a nation weakens and its standard of living declines, it introduces the possibility of total governmental collapse. The Fund for Peace characterizes a failed state as having the following characteristics:
Loss of control of its territory, or of the monopoly on the legitimate use of physical force
Erosion of legitimate authority to make collective decisions
Inability to provide public services
Inability to interact with other states as a full member of the international community
Common characteristics of a failing state include a central government so weak or ineffective that it has an inability to raise taxes or other support and has little practical control over much of its territory and hence there is a non-provision of public services. When this happens, widespread corruption and criminality, the intervention of state and non-state actors, the appearance of refugees and the involuntary movement of populations, sharp economic decline, and military intervention from both within and without the state in question can occur.
Metrics have been developed to describe the level of governance of states. The precise level of government control required to avoid being considered a failed state varies considerably amongst authorities. Furthermore, the declaration that a state has "failed" is generally controversial and, when made authoritatively, may carry significant geopolitical consequences.
Now for the violence part:
According to the political theories of Max Weber, a state is defined as maintaining a monopoly on the legitimate use of physical force within its borders. When this is broken (e.g., through the dominant presence of warlords, paramilitary groups, corrupt policing, armed gangs, or terrorism), the very existence of the state becomes dubious, and the state becomes a failed state. The difficulty of determining whether a government maintains "a monopoly on the legitimate use of force", which includes the problems of the definition of "legitimate", means it is not clear precisely when a state can be said to have "failed".
The problem of legitimacy can be solved by understanding what Weber intended by it. Weber explains that only the state has the means of production necessary for physical violence. This means that the state does not require legitimacy for achieving a monopoly on having the means of violence (de facto), but will need one if it needs to use it (de jure).
You can read the rest here: https://en.wikipedia.org/wiki/Failed_state