The Cold War developed out of post World War II rivalries between the United States and the Soviet Union. With the destruction of Germany and the weakening of Britain and France, the USA and USSR were the only two powers remaining. The conflict that would become the Cold War emerged from a dispute over what would become of Eastern Europe after the war. Franklin Roosevelt and Stalin had vaguely agreed at the wartime conference at Yalta that Poland would have free elections and a democratic government. Eventually, though, it became clear that Stalin would establish a puppet communist regime there. By the time Truman and Stalin met at Potsdam, the United States was convinced that the Soviets were attempting to conquer all of Eastern Europe. Stalin, for his part, was concerned with ensuring that the Soviet Western border was secure, and saw American plans for the reconstruction of Western Europe as equally imperialist in nature.

This belief was intensified with American support for anti-communist fighters in Greece and Turkey and eventually the massive aid package for western Europe known as the Marshall Plan. By 1948, the two allies found themselves facing off across a divided Germany and elsewhere along what Winston Churchill called an "iron curtain" that stretched across eastern Europe. In the meantime, China had become a communist nation under Mao, and North Korea would soon, with Soviet support, invade the south. Both sides came to see all of geopolitics in terms of a struggle between communism and western democratic capitalism.

How did the Cold War develop in the Soviet Union, Europe, and the US?

If you are seeking to answer this question from what has been discussed in a classroom setting with instructor or through a class text, these resources should serve as your primary basis.  Outside of that, the fear of "the other" that both sides possessed in healthy doses helped to develop the Cold War.  Unified in their mutual dislike and threat in Nazi Germany, the United States and Soviet Union put aside their disagreements and ended up collaborating for the good of putting down the expansionist threat of Hitler.  Once the war ended, both sides continued with their mistrust of one another, heightened by the American use of the atomic bomb and increased with the Russian launching of the Sputnik space satellite.  In the end, the fear of the other helped to erect divisions in Europe where sides were chosen.  Sometimes, this choice was made through political convenience in siding with the other and other times it was made completely through being forced by "the other."  This fighting for supremacy allowed the Cold War to spread.

Your book surely has specific facts you are supposed to come up with because this is such a broad question.  You really should check your text to make sure you get the right facts.

The Cold War developed as a competition between the Soviet Union and the United States.  Much of the early Cold War was played out in Europe.  The Cold War started with the partitioning of Europe at the end of World War II.  From there, the positions of the two sides became hardened as the two superpowers became more and more suspicious of each other.

There was some amount of a push in various European countries to separate themselves from the Cold War competition.  The French tried to assert themselves as an independent power.  On the communist side, so did Yugoslavia.  But overall, the pattern held -- the world order in the Cold War was very much one of the US and its allies against the Soviets and their client states.