A state could be said to "succeed" if it maintains a monopoly on the legitimate use of physical force within its borders. When this is broken, the very existence of the state becomes dubious, and the state becomes a failed state. The difficulty of determining whether a government maintains a "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 controversy derives from the political and military implications of labelling a state as "failed". The proclamations and laws of its government may be ignored, and in some cases violent action may be undertaken inside the borders of the "failed state" by agents from other countries; such action naturally has highly dubious legality.
Western politicians have labelled many countries as failed states, including: Afghanistan (under the Taliban), Somalia, Yemen, Georgia.
See also: rogue state
Red Cross article: The "failed State" and international law by Daniel Thürer