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.