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

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.