The arms race made the Cold War more tense and heated.
An example would be the Cuban Missile Crisis in 1962. The United States had placed missiles in Turkey, which the Soviet Union saw as a threat. The USSR then sought to place missiles in Cuba, its newly communist ally in the Western hemisphere. It is often stated that this moment was the closest the USA and the USSR ever came to the eruption of an actual nuclear war between the two superpowers.
As the Cold War continued, the two superpowers (US and USSR) kept escalating their weapons capabilities and stockpiles. It got to the point that if the two sides did plunge into war, they would face mutually assured destruction. Eventually the US and USSR would pursue policies of detente, which included pledges to reduce their nuclear arsenals. The arms race and solving the arms race were constant issues affecting the Cold War.