The term "Cold War" was first used in the post-WWII era by George Orwell. He used the term in a generic way to describe any state of perpetual conflict between two antagonists. He did not specifically apply it to the relationship between the US and the USSR.
The first use of the term to specifically refer to the conflict between the two superpowers is believed to have been by Bernard Baruch. He had been an important advisor to many US presidents, particularly Franklin Roosevelt and Harry Truman. Baruch first used the term in a speech on April 16, 1947. After Baruch's use of the term, it was picked up by others, including Walter Lippmann, who wrote a book entitled The Cold War. Since Lippmann was much more in the public eye than Baruch was, his use of the term is seen as a major factor in causing it to be widely used.