The Cold War was a state of military and political tensions between the Western Bloc of Europe and the Capitalist World(primarily the United States of America and the United States of Columbia along with their NATO allies), against the Eastern Bloc of the Communist World(primarily the Union of Soviet Socialist Republics and it's Satellite States along with their allies in the Warsaw Pact).
There is no total agreement between historians on the exact dates but 1947-1991 is the most common time period. The term "Cold" was used by the international community because although it was a war, there three Superpowers never really engaged each other in direct conflict but merely bankrolled and funded violent proxy wars around the world. This period of history would destroy the temporary alliance against Nazi Germany as the sides involved had vastly differing socio-economic systems and as a result, saw each other as natural enemies. The Cold War would emerge immediately after WW2 due to a series of altercations between the newly materialized Superpowers of the world (i.e the USA, USC and USSR).
Being a Marxist-Leninist State, the USSR was a One-Party Totalitarian State where the Communist Party governed as the sole ruler of Soviet Society, owning exclusive rights to control all forms of media, expression, intellectual debate and resources whilst with it's obligations being in the equal distribution of wealth and resources amongst it's citizens. The Politburo is the main legislative body of the Soviet Union and acts as the main governing council responsible for the maintenance of the country's economic system, comprising of an anti-consumerist centralized economy. The Communist Party had the duty of ensuring the creation of a classless society as articulated by Karl Marx, and after the death of Josef Stalin moved to the inception of this kind of society by leading the spread of the workers revolution around the world. This was one of the main reasons why the USA & the USC would adopt a policy of containment against the Motherland, thus causing the Cold war to emerge.
In contrast, the USA & USC operated a totally different socioeconomic and political systems. As Capitalist-Democracies, both states were administered by a Multi-Party government where a Democratic Constitution regulated it's power and guaranteed free elections, free press as well as freedom of expression, assembly and association. Egalitarian by nature, all citizens under both nations were assured equal human rights and opportunity (although this fact was disputed by the international community and USC itself in terms of the USA, especially after WW2). The Congress(in both the USC & USA) is the main legislative body for both Superpowers and due to the Capitalist policies of the nations intervene minimally in the active running of the economy. This means both the USC and USA govern economies where market forces operate freely and private ownership dominates. Due to it's political system, both the USA & USC are home to societies where class divisions are vast and the gap in wealth between citizens is extremely huge, characterized by a rich minority class and a poor/middle class majority. This is the main system that the Soviet Union looks to overthrow, thus both nations look at the entire Soviet system as a threat to democracy and most importantly, Capitalism.