Short answers, pulled from the story.
The nuclear arms race began during World War II, rooted in competition over the German nuclear program and its materials. The US-Soviet dimension accelerated after the atomic bombings of Hiroshima and Nagasaki in August 1945, when the Soviet Union intensified its own atomic bomb project, leading to the first Soviet nuclear test on the 29th of August 1949.
The Tsar Bomba was a Soviet hydrogen bomb detonated on the 30th of October 1961, with a yield of approximately 58 megatons. It remains the most powerful nuclear device ever exploded and marked the peak of atmospheric nuclear testing before the 1963 Limited Test Ban Treaty.
During the Cuban Missile Crisis in October 1962, the United States raised its military forces to DEFCON 2, the highest alert level reached during the Cold War. Soviet First Secretary Khrushchev and President Kennedy communicated directly to defuse the crisis; on the 28th of October 1962, Khrushchev announced Soviet missiles would be withdrawn from Cuba.
Klaus Fuchs and Theodore Hall provided the Soviet Union with detailed designs of both the implosion bomb and the hydrogen bomb while working inside the Manhattan Project. Fuchs was arrested in 1950, which led to the arrests of Harry Gold, David Greenglass, and Ethel and Julius Rosenberg, the latter two of whom were executed for espionage in 1951.
Mutual assured destruction, known as MAD, was the strategic doctrine under which both the United States and the Soviet Union maintained enough nuclear capability to survive a first strike and still deliver a devastating retaliatory blow. By the mid-1960s, both sides had developed this second-strike capability, creating a situation where initiating nuclear war would guarantee the attacker's own destruction.
The Federation of American Scientists estimated in 2015 that 125,000 nuclear weapons were produced worldwide between 1945 and 2013. The United States and Soviet Union accounted for the vast majority of that total, with both countries' arsenals eventually holding thousands of warheads each.