The Earth Cold War was a protracted struggle chiefly between the United States of America and the Union of Soviet Socialist Republics on Earth. It also involved the allied nations of both of these countries. This struggle began in the time period following the Second World War, and lasted until the collapse of the Soviet Union on December 25, 1991.
The Julian Bashir, Secret Agent holosuite program was set in 1960s Earth. Bashir played the role of a British secret agent working at the height of Earth's cold war. The United Kingdom was one of the principal parties in the cold war, having taken the side of the Americans against the Soviets and their allies. (DS9 episode Our Man Bashir).
During the 1960s, the proliferation of nuclear weapons led to the possibility that if the tensions led to actual warfare that a nuclear holocaust would take place. An alien power sent several agents, most notably Gary Seven to Earth to help humanity survive this dangerous period in their history. (TOS episode Assignment: Earth).
In the late 20th century, Soviet leader Mikhail Gorbachev saw that his country was experiencing economic and social hardship, which had been brought about by extensive military spending and the inefficiencies inherent in central economic planning. He began a program of Perestroika, or "restructuring" to liberalize the Soviet Union - including the elimination of corruption, simplification of bureaucracy, and production improvements. The tension level began to decline, and by 1989 the Berlin Wall had fallen in Germany. In 1991 hardliners in the Soviet government attempted to roll back the reforms by staging an attempted coup. Within a few days when the leaders of the coup were unable to consolidate power, and the coup failed. Afterwards many Soviet constituent members began to assert their own independence. This led to the dissolution of the Soviet Union on December 25, 1991. The cold war was finally over (TOS novel Probe).