The "Cold War" was period of strained diplomatic relations between the Soviet Union (USSR), a communist-governed nation, and an organization called NATO (North Atlantic Treaty Organization) beginning roughly at the end of World War II. Relations, especially between the USSR and the United States (USA), were very tense, a feeling that was exacerbated by the presence of nuclear weapons the USSR and USA had built and aimed at each other during the "arms" race following the use of the world's first atomic bombs by the USA on Japan at the end of World War II. Both nations felt that nuclear war was a very real threat and sought to out-do each other by producing as many atomic weapons as possible.
It was during the early part of the 1950s that Senator Joe McCarthy of Wisconsin fueled public tensions by announcing that communists and others suspected of sympathizing or allying themselves with communists were operating within the borders of the USA. McCarthy and his followers began publicly accusing individuals of being communist without proof. A distant parallel to this event can be seen in the efforts of Senator Keene to ban costumed adventuring.
Also taking place during the early 1950s was the Korean War, involving the United States and the communist contingency that attempted to seize control of Korea. The result was an eventual treaty with no real victor; the country was divided into two nations at the 38th degree of latitude, where a "demilitarized zone" (DMZ) was established that neither nation would violate.
The Vietnam War was also a byproduct of the Cold War. United States government and military officials believed that the takeover of Vietnam by Ho Chi Minh, the communist party leader of that Southeastern Asian country, would create a "domino" effect that would lead to the communist seizure of more nations of the world. The Comedian and Dr. Manhattan intervenes in the Vietnam War and brings it to an end with the USA victorious.
Adrian Veidt's murder of two million New York City residents when he teleports a giant, alien-appearing creature there, its psychic shockwave killing anyone near to the epicenter of the event.
The actual invasion in Afghanistan took place in 1979 but it seems that in the graphic novel Dr. Manhattan's presence stopped the Soviets. The invasion lead to a 10 year war that the USSR eventually lost, with mostly Islamic-inspired Afghan "mujahideen" tribes of the country uniting under U.S. military support to fight the USSR.
As a result the Cold War ended with the dissolution of the Soviet Union because the war, among other factors, drained the Soviet finances and also due to lengthy diplomatic negotiations between the USSR's last standing president, Mikhail Gorbachev and the U.S. president of the time, Ronald Reagan, culminating in the official declaration of the end of the Cold War in 1989.