The term is defined in FAA Order JO 1900.47, Air Traffic Control Operational Contingency Plans.
Examples of events that might result in degraded operations include technical system failures, errant construction, hurricanes, tornadoes, floods, wild fires, earthquakes, pandemic, terrorism, or war.
The TFR would state what aircraft are permitted into the affected airspace (e.g., military, first responders, fire fighting, surveillance) and under what constraints and requirements those flights can operate.
Aircraft already in the affected airspace would transition to VFR procedures, maintain their last clearance as able (trajectory, altitude, airspeed, heading), attempt to establish communication with other aircraft or ground assets to effect procedural self-separation, and then as able may decide to land or continue to an alternate outside the affected airspace.
On August 25, 2018, Miami International Airport (MIA) declared ATC Zero following a power outage at the Miami TRACON[3][4] On December 15, 2021, Kansas City International Airport (MCI) declared ATC Zero for approximately one hour when a winter storm brought 80+ MPH gusts prompting a rare evacuation of the tower which at the time was 265’ high.