The Cold War had many effects on the two great powers of the United States and the Soviet Union and those effects were different at different points in the Cold War. I will discuss what I see as the two main effects that the Cold War had overall.
On the United States, I would argue that the Cold War's main effect was a willingness to go against American values in foreign policy. The US was consistently willing to back dictators and other oppressive regimes in the "Third World" so long as those regimes were anti-communist. Some examples include the apartheid government of South Africa, military dictatorships in South Korea and Ferdinand Marcos in the Philippines.
For the Soviet Union, the main effect was that the Cold War destroyed their country. The pressure put on the USSR, both military and political pressure, caused tensions to rise in the country. The Soviet government was forced to use its resources for military equipment instead of for increasing its people's standard of living. It was forced to be very repressive for fear that openness would cause it to lose people to the West. These things made people more and more unhappy to the point that they exploded as the Cold War came to a close.
These are, as I see it, the most important effects that the Cold War had on each of the major powers.