A society that has no government would be considered an anarchy. Government mediates disputes between people, provides basic services like roads, education and national security and enforces laws to discourage wrongdoing from being committed. All of these would not exist in a true anarchy.
Nations without government have not truly existed in modern history. However, there have been instances of anarchy occurring in countries undergoing abrupt and sometimes violent changes. For example, France after the French Revolution temporarily did not have a government. More recently, Somalia was in a state of anarchy following the collapse of its government, and legal affairs were handled by local communities according to tradition and religious law.
Countries that have experienced collapse of their government are often referred to as "failed states." This creates a power vacuum; the natural inclination is for people to seek power, and various factions often fight for control, creating chaos and violence. Additionally, there are no protections for citizens from crime, no legal remedies and no basic services. In 2013, Lebanon was considered to be at risk of becoming a failed state. The Syrian crisis in the region destabilized the nation, causing sectarian violence and diminishing the influence and authority of the government.