Discuss how the Cold War between the United States and the Soviet Union affected other nations and/or regions of the world.

Whether they liked it or not, most of the world's countries had to take sides during the Cold War. The United States, then as now, was the world's biggest superpower, with the world's largest economy. It was virtually impossible for any country to avoid having to deal with the United States; the only question was how. Countries such as India adopted what could be described as a flexible posture with regards to the Cold War, playing the United States and the Soviet Union off against each other, depending on the circumstances.
In the developing world, both new nations and anti-colonialist movements alike came under the influence of the USSR, which was seen as a useful ally against Western imperialism. For its part, the United States also involved itself in the affairs of the developing world, often propping up dictatorships that were hostile to Communism. Continents such as Africa and South America became battlegrounds in this epic ideological struggle, with both the Americans and the Soviets jockeying for strategic primacy in advancing their respective goals.
As the United States and the Soviet Union were nuclear powers, a direct confrontation between them was unthinkable. So the two Cold War superpowers used proxies to fight their battles for them. A prime example of this can be seen in the Angolan Civil War, where two rival liberation movements—the Marxist MPLA (backed by the Soviets) and the more right-wing UNITA (backed by the United States)—engaged in a long, bloody and bitter struggle for nigh on thirty years. The war was prolonged by massive shipments of arms to both sides in the conflict from their respective Cold War backers.
Many Angolans simply wanted independence; they weren't too concerned about the wider geopolitical issues involved. Nonetheless, they found themselves caught up in the middle of another conflict which caused significant suffering and misery to their country. What happened in Angola is symptomatic of how the world was divided up during the Cold War and of how some kind of involvement was almost inescapable, especially for the world's poorest and least-developed countries.

The Cold War affected the entire world, not just members of the two nations primarily involved.  The United States and the Soviet Union fought a series of proxy wars in the developing world.  Both the Soviet Union and the United States initially sent advisers to North and South Vietnam respectively before moving to send combat forces, the United States sending ground troops and the Soviet Union sending fighter pilots.  The United States and the Soviet Union also fought a proxy war in Afghanistan, with the Soviets sending ground troops and the United States arming the mujaheddin rebels.  Both of these proxy wars served to destabilize both countries.  Millions of Vietnamese, Laotians, and Cambodians died as a result of the Vietnam War, and the world is still experiencing the results of extremism as a result of the chaos that was created in Afghanistan during the 1970s.  
Even regions that did not experience war directly were affected by the Cold War.  Germany remained divided between East and West well after World War II was over.  Even today, American troops are a presence in Germany.  Cuba received Soviet aid for decades; this nearly led to the island being invaded in 1962 and has led to sanctions that has affected all Cubans.  Many armies  and paramilitary groups around the world still use Soviet-era weapons.