The fact that the Soviet Union and United States had atomic weapons that they could use against each other greatly affected the Cold War. In the United States, students in schools learned about what to do in case of a nuclear attack. The popular "duck and cover" method was introduced to America as a means to protect themselves if atomic warfare ensued.
As far as the relationship between the US and Soviet Union, they were constantly worried about if/when the other country would use this type of weapon. This resulted in both countries investing hundreds of millions of dollars into their military complexes and space programs. Ultimately, this competition over weapons and the constant fear of attack lead to the concept known as brinkmanship. This refers to the fact that the United States and Soviet Union were constantly on the brink of all out war, which resulted in the deaths of millions of people all over the world.