The Cold War was the name given to the bipolar division of the world between the US and the USSR after the end of World War II, when these countries, strengthened by winning the war, disputed the economic hegemony of the world. The US sought to consolidate the capitalist model of production and the USSR sought to implement socialism.
The war itself never happened, it was just a war of influence. If an armed conflict had occurred, the US would have support from the already capitalist nations such as France, Canada, Brazil, England, Australia, Italy, etc. In the case of the USSR, the aid would come from communist countries such as North Korea, Cuba, China, and the USSR countries themselves.