Why was the Cold War so influential in shaping the modern world?
That's a great question. The Cold War has left a long lasting imprint on the world as a whole, and certain regions and countries in particular. Part of the reason is because the world became so polarized around either the West or the East, and the contrast between the two sides was so clear. A second reason is that the Cold War was in fact a world war, as it was waged on every continent, and over a long period of time. I'll start with the legacy in the United States and the USSR, the two main competitors in that war.
The United States: At one point, the US had nearly 12,000 nuclear weapons, and today still has nearly 4000. Hundreds of billions of dollars were sunk into America's nuclear program and military between 1945 and 1991, and represent a significant chunk of America's debt today. Even today, 33,000 US troops are in South Korea, 150,000 in European countries, Cuba is under trade embargo, and Russia and the United States are often adversaries on the world stage. The word "socialism" is a cuss word in American politics because Americans, especially the older generation, had been so sensitized to it during the Cold War.
The USSR: The Cold War, among other things, is what led to the disintegration of the Soviet Union in 1991, which profoundly shaped the current map and political situation in the former 15 Soviet republics. Russia still has thousands of nuclear weapons, massive stockpiles of military equipment, and competes with the West for influence in the world.
Africa: Unfortunately, Africa got the worst of the Cold War in my opinion. After colonialism faded in the 1960s, the US and USSR stepped into the vacuum, providing billions in aid for dictatorships that promised loyalty and flooding the continent with modern weapons. We can see the long term legacy of those policies in Africa's long-running conflicts today.