Personally, although it often got boiled down in popular consciousness to "communism", the big, primary concern of the US government was Soviet influence. Which is to say, for example, while a lot of Americans in the 1950s would look askance at any sort of Marxist group, the big concern among the government and authorities were groups that were either openly favorable to Soviet foreign policy (like CPUSA) or groups that might be front organizations controlled by Communist Party members, or individuals under the influence of Soviet intelligence. This was a major driver of the Red Scare of the late 1940s and 1950s, although as I wrote here it was singularly ineffective in finding Soviet spies.

Another reason that "fighting communism", while it might have made a good slogan, does not accurately reflect US policy in the Cold War are the times the US developed friendly ties with communist regimes. The most famous will of course be the People's Republic of China, which the US developed favorable relations with from the early 1970s on. This was clearly for geopolitical, not ideological reasons. Nixon made a political career of red-baiting, while Mao had long denounced the USSR as "revisionist" and traitors to Stalin's legacy, yet both men met and helped develop cordial relations between both countries (the US would sell billions of dollars' worth of weapons to the PRC until 1989).

And China was not the only example. Tito's Yugoslavia, after breaking with Stalin in 1948, developed warmer ties with the US, even winning foreign aid. Ceausescu's Romania, pursuing an independent foreign policy (it did not participate in and criticized the Warsaw Pact's invasion of Czechoslovakia in 1968), also developed good ties with the US, with the government promoting private US loans to the country and Ceausescu even getting a photo op with the First Family from a White House balcony.

Support for other communist regimes or movements that were fighting the Soviets or Soviet allies was often indirect or clandestine, but very much a reality. Official recognition of the Chinese-client Khmer Rouge as the official Cambodian government after the 1979 Vietnamese invasion is perhaps the most notorious example, but there are a few others as well, such as US support for Siad Barre's Somalia after it went to war with Ethiopia (Somalia and Ethiopia were both Soviet clients at the start of the war, but the Soviets threw in behind Ethiopia, so Somalia eventually got US support). Angola's UNITA would likewise be lionized as anti-communist freedom fighters in the 1980s, but had originally started out as a PRC-funded communist group before the US took over financing it.

The answer by u/MAXSquid does get at some important themes, but I think there is much more to the history of Communism and anti-Communism in the United States that we can delve into.

The First Red Scare in the United States comes about after the Bolshevik Revolution in Russia in 1917. Not only was Bolshevism the antithesis of the liberal capitalist tradition in the United States, but this was also the height of wartime anxiety, suspicion, and jingoism in the United States. I've written here about the surveillance programs that got their start during WW1. Communism was linked to subversive activity, anti-state movements, and anarchism, all of which were directly counter to the interests of the US state. So from the first moment that Communism as a political force really has a large presence in the mindset of most Americans, it is linked to the notion of subversion or treason.

Communism reemerges as a boogieman in the aftermath of WW2. I'm sure there are countless answers on the suspicion between the Allied Powers and the quick collapse of the old wartime alliance. To put it simply, the Truman administration soon agreed with Churchill's assessment that the Soviet Union was aggressively expanding its sphere of influence and forcibly installing Communist governments in Eastern Europe as opposed to letting those formerly Nazi-occupied states go their own way. Much like how electoral victories of Fascist parties led to the end of democratic elections during the 1930s, there was a similar concern that Communist governments emerging in European countries would mean their automatic allegiance to Moscow and authoritarianism. There was also geopolitical rivalry for very material reasons, and the Western allies rightfully were concerned that Europe would fall entirely under the sway of the Soviet Union. Thus communism was hitched to rivalry with the Soviet Union.

Building off of wartime narratives about the importance of loyalty (and the accepted notion that most Japanese-Americans could not be trusted to be loyal), the moment was ripe for increased public and government anxiety about the possibility of disloyal Communist subversives within the US. Part of the reason why the US government tended to see subversion as a concern, as opposed to outright Soviet aggression, was precisely because the Truman administration was confident that the Soviet Union would need several years to rebuild its strength after WW2. The logic here was that while the US had the advantage in most areas of conventional strength, the Soviet Union would resort to subversive activities and espionage in order to either weaken the United States or close the technological gap through theft.

By the time Senator Joseph McCarthy begins making waves in Washington, through introducing internal security acts, alleging the presence of disloyal communists within the ranks of the Truman administration, and calling for the termination of high-ranking US officials, the stage was set for broad public acceptance of the narrative that Communist spies could be anywhere.

Why is anti-communism invoked to justify US interventions abroad? Going along with the assumption that communist movements are inherently linked, in some fashion, to Moscow, the activities of Communism globally becomes linked to foreign policy concerns. Thus, the Korean War was not just a civil war story of the communist North attacking the South--it was a story of Moscow advancing its geopolitical position in the unstable East Asian region. The victory of Chinese communists in 1949 against the better-equipped Nationalist regime in China, also, increased American anxiety about how Communist movements could take root among the rural poor and take over a country, further turning the tides against the US.

Was there ever a chance for communism to actually be a domestic threat in the United States? Probably not. But the shocking victory of Communism in China, the rapid installation of Communist regimes in Eastern Europe, and the perception that it was very, very easy for Communist agitators to build a movement among the poor and destitute, added to an American sense of vulnerability.

And yes, anti-communism and Red Scares happen all over the world during the 1950s. Masuda Hajimu has a fascinating book which explores the series of anti-leftist crackdowns that happen in Taiwan (the White Terror), Great Britain, and Japan. While some crackdowns were indeed surface-level justifications for certain political elements to advance their own interests, the anti-Communist justification was an easy one precisely because of the overwhelming sense of anxiety and fear of subversion and instability in the postwar period.

Sources: Uncle Sam Wants You by Chris Capozzola for WW1 Red Scare, Masuda Hajimu's Cold War Crucible for the 1950s moment of anti-leftism worldwide, The Marshall Plan by Benn Steil for US govt logic behind pouring foreign aid into Europe, and A Preponderance of Power by Melvyn Leffler for a blow by blow account of the Truman administration's policies.

Greetings! I shall build off the excellent answers of u/Kochevnik81 and u/CapriciousCupofTea by delving a bit more into the geopolitical side of the Cold War's origins. It remains of course, a historiographical debate of considerable (though by no means dominant or overarching) influence as to what extent ideology really motivated the US to do what it did over those 50 or so odd years. Our starting point for this is 1945, Ian Buruma's cheerfully named "Year Zero". Let's begin.

The Potsdam Problems

As Kochevnik81 pointed out in their response, Communism may have served as the ideological byword of "Soviet influence". The United States did not, as CapriciousCupOfTea notes, exactly find the Bolshevik revolution in Russia and the resulting government of the USSR "to their taste", so to speak. The Second World War of course, put aside those ideological differences for the time being, as both Uncle Sam and the Soviet Sickle had a common enemy to defeat: Hitler's Nazi Germany. Once that threat had been defeated after four years of taxing and brutal conflict however, it remained unclear as to what the USSR and USA had in store for Europe. The British Empire was now a setting Empire, so historians often point to the post-war period as the emergence (but not necessarily a rapid one), of the "two-superpower world".

The first signs of tension between the two superpowers came during the Potsdam Conference (July 17th - August 2nd, 1945). Joseph Stalin was the last of the "Big Three" Allied leaders who remained in power (Churchill lost the general election to Attlee during the conference, and Truman had replaced the late President Franklin D. Roosevelt after his death in April). This sparked misunderstandings and a fair bit of concern from both sides. Truman for his part, was not at all experienced with the Soviets, and he was more suspecting of their intentions than Roosevelt had been. He even recorded after a meeting with Stalin at Potsdam that "the Russian were planning world conquest".

The key question on everyone's mind was the shape of post-war Europe, and it was here that we get the "spheres of influence" dimension to the whole geopolitical mess that would become the Cold War. Stalin, with the cooperation of Roosevelt and Churchill, believed that the Soviet Union deserved the rights to exercise some influence (but not control, mind you) over Eastern Europe (Poland, Czechoslovakia, Romania, Hungary, Bulgaria etc.). Truman however, alongside his Secretary of State James F. Byrnes, believed (perhaps rightly so) that Stalin would immediately begin transforming these nations into Soviet puppet-states. Byrnes was particularly concerned with Poland, where Moscow had set up a sponsored government in January called the "Lublin Poles", which complicated the fact that the pre-war Polish government-in-exile (the so called "London Poles") also wished to govern the country. In the end Byrnes agreed to give recognition to the Soviet government, but insisted (as Churchill had at Yalta), that Stalin allow "free elections" (a term which pops up a lot in American diplomacy during the Cold War, and in some instances a hilarious irony) to take place at the earliest possible moment. Stalin viewed this insistence as signs of American influence, which he found somewhat concerning.

For the rest of Eastern Europe, hard facts dictated the options. The Red Army had liberated much of the region from Third Reich, so Stalin had practically incontestable reason to exert his influence over these states. Historian Thomas Patterson notes however, that this influence was itself motivated due to "profound security fears" over Eastern Europe, as it had been from this region that the Nazis had launched their catastrophically bloody war against the Soviets.

The Long Telegram

The concerns which Truman left Potsdam with were further reinforced in February 1946, when George F. Kennan, head of the US Embassy in Moscow, telegrams a 5,653 word report on the state of Soviet intentions with regards to the US and Europe. Here's an extract from that telegram:

“We have here a political force committed fanatically to the belief that with the US there can be no permanent modus vivendi, that it is desirable and necessary that... the international authority of our state be broken, if Soviet power is to be secure... Many foreign peoples, in Europe at least, are tired and frightened by experiences of the past, and are less interested in abstract freedom than in security. They are seeking guidance rather than responsibilities. We should be better able than the Russians to give them this. And unless we do, the Russians certainly will.

In July, Truman himself ordered adviser Clark Clifford to write a top-secret report based on the "Long Telegram" in order to form a new and concrete American foreign policy towards the USSR. The result, the Clifford-Elsey report, was instrumental in the Truman Doctrine to follow (and the ideological spin which it evolved):

“As long as the Soviet Government adheres to its present policy, the United States should maintain military forces powerful enough to restrain the Soviet Union and confine Soviet influence to its present area. All nations not now within the Soviet sphere should be given generous economic assistance and political support in their opposition to Soviet penetration.”

Note that the Clifford-Elsey Report and the Long Telegram did not actually frame Communism as the main reason for Soviet expansionist desires, but both reports considered it a possible reason for Stalin's "aggressive" designs for Eastern Europe. The Soviets for their part, eyed the American "Open Door" Policy (by which American economic and even military influence was spread across the world, sometimes under force) with suspicion, fearful that such influence would undermine their "bloc" in Eastern Europe.

The Truman Doctrine

On March 12, 1947, Truman presented his new policy to the American Congress. He took care not to explicitly mention the Soviet Union, but he did emphasise (and imply strongly) the aspect of Communism in fueling Soviet doctrine, something which key advisers (among them Kennan himself and Secretary of State Dean Acheson) would later critique:

“I believe it must be the policy of the United States to support free peoples who are resisting attempted subjugation by armed minorities or by outside pressures...I believe that we must assist free peoples to work out their own destinies in their own way... I believe that our help should be primarily through economic and financial aid which is essential to economic stability and orderly political processes.”

What followed was the Marshall Plan, which further incited Soviet concern over America's increasing economic and political hold over Western Europe. To that end, in September of 1947 Stalin and the heads of various Communist parties across Eastern (and even Western) Europe met in Poland to form the Information Bureau of the Communist and Workers' Parties (shortened to "Cominform"). At the conference where this happened, Andrei Zhdanov gave his famous "Two Camps Speech", now solidifying the ideological and geopolitical rivalry which had come about:

“A new alignment of political forces has arisen. The more the war recedes into the past, the more distinct become two major trends in post-war international policy, corresponding to the division of the political forces operating on the international arena into two major camps: the imperialist and anti-democratic camp [America], on the one hand, and the anti-imperialist and democratic camp on the other [The USSR]..The vague and deliberately guarded formulations of the Marshall Plan amount in essence to a scheme to create blocs of states bound by obligations to the United States, and to grant American credit to European countries as a recompense for their renunciation of economic, and then of political, independence."

I don't see where any of the answers addresses the threat that the spread of Communism would reduce the profits of US Capitalists by limiting their markets, investment opportunities, and access to cheap labor and natural resources.

Were these factors unimportant?