The World Wars had exhausted the power and influence of the contemporary economic and military empires around that time and America and the Soviets emerged as the leading superpowers in terms of technology and economy around that time. This lead to the polarization of the world in two parts. The conflict arose as a result of difference on opinions regarding the future of the post world war era as the conflict of two major economic ideologies namely Capitalism and Lenin's version of Communism. Being nuclear powers possessing Weapons of Mass Destruction (which served as a deterrent to a direct conflict) it was not fought as a conventional war but both powers employed vast resources and fought several proxy wars and used lesser developed nations as pawns in their conflict. It lead to the arms race, the space race and a race to establish hegemony over the rest of the world. Besides that, they were involved in espionage and sabotaging each others influence on different fronts. The cold war finally ended with the dismemberment of the Soviet Empire during the Afghan conflict in the late 1980s.
It was fought threw espionage, propaganda, and proxy armies. Never fought by militaries thought.
The cold war was never fought.