The two opposing alliances were the Western bloc (USA and its NATO allies) and the Eastern bloc (Russia and its allies in the Warsaw pact). The Cold War left the USA and USSR the two superpowers of the world that have great economic and political differences.
They never really did engage in a full-scale armed combat but both of them were prepared for a possible all-out nuclear world war. Each side had its own nuclear deterrent and would imply that the attack of the other means the destruction of the other as well. This was known in history as the doctrine of Mutually Assured Destruction (MAD). Massive propaganda campaigns, espionage, and psychological warfare were dominant during the cold war between the two superpowers.