The popular fear of communism was rooted in the legacy of repression and violence among self-described communist regimes. While some would assert that the Soviet Union, the Khmer Rouge, the DPRK, Maoist China, etc. did not represent "real" communism, those regimes (and their crimes) were clearly associated with communist ideology in the popular imagination. The prospect of such a regime rising in one's own country was understandably worrisome.

When one looks at the issue economically it comes down to who controls the flow of capital in the economy. In capitalism that flow is driven for the most part by private individuals. In a communist/ socialist economy the government takes a more prominent role in the movement and allocation of capital. This is where the debate takes place between economists that continues to this day.

What I definitely want to address is the public perception of what Communism or Socialism is. First of all one mistake people make is equating Communism, specifically what was practiced in the Soviet Union, as Marxism. This is not true. While Karl Marx was influential to the leaders of the Bolshevik Party and the subsequent Communist government in the USSR, he himself didn't write the Communist Manifesto as an economic model that should be pursued by society. He wrote it as a critical analysis of the Capitalist system which had risen to prominence in conjunction with the industrial revolution. He surmised that as a result of the exploitation by the wealthy class, for whom capitalism provided an optimal environment for success, the working class would rise up in revolt and try to establish a state that served their own interests.

As for the rivalry between the two systems throughout the 20th century focused on the U.S. and the USSR. It was an economic competition for resources and hegemony throughout the world as both sides sought to exploit foreign markets while at the same time combining an arms race of unprecedented proportions which of course raised the stakes and the tension when standoffs occurred (think the Berlin Crisis or the Cuban Missile Crisis).

To backtrack a little bit, the ultimate example of Communism is the Soviet Union so I'll discuss them and the US a little more. As you mentioned the US and UK supported the Royalists against the Bolsheviks. Naturally this led to distrust between the two which has snowballed the years and become entrenched within each political faction's platform.

Ultimately at the end of the day it comes down to economics. You mention the fear of losing power and wealth by capitalists and business magnates and this is very true. One can just look within American politics today and see the way the likes of the Koch brothers behave with regards to campaign finance. Also with how once President Obama started to push for raising the taxes back to the Clinton era levels and then his affordable care Act, immediately people on the right started to label him as a socialist even though he is far from a socialist (his healthcare plan is basically a republican plan similar to Bob Dole's). This competition between big businessmen and the workers is not new. One can even go back to Ancient Rome and look at the efforts by the Patrician class against the Gracchus brothers and later the vilification of Catiline and the assassination of Caesar who were men who acted in the interests of the plebeian class specifically with regards to citizenship and land redistribution.

So the stigma towards Communism or socialism and sometimes it's called populism largely comes from economic desires. It is in the interests of private businesses to have the state play as small a role in the economy as possible so naturally those who fall under the capitalist persuasion will be opposed to a socialist movement.

Communism (if by "Communism" you mean Marxism and its derivatives, as most people do) is a revolutionary political ideology, so by definition, its adherents pose a direct challenge and threat to forces of power and authority (i.e. the "status quo") in liberal, bourgeois, capitalist societies.

If you look at the social backgrounds of those attracted to socialist, communist, anarchist, and other radical left-wing movements in the United States at the turn of the last century, for example, they tended to be poor workers from Southern and Eastern European immigrant backgrounds. Lots of Jews, Catholics, and Orthodox represented.

When you combined these immigrants and their non-Protestant, foreign religions with their marginalized status as poor working men and women, it was all too easy for native-stock, white Protestant Americans to equate socialism with foreign, un-American, "subversive" viewpoints, and vice versa. It's no coincidence that the First Red Scare occurred during a period when massive waves of poor, under-educated immigrants from southern and eastern Europe were coming to the US (in addition to the Communist Revolutions occurring in Europe at the time).

On top of all of that, America's involvement in the First World War served as a pretext for the US government to crush the domestic socialist and other radical movements, all in the name of "Americanism" and "patriotism." Many Americans were deeply fearful and suspicious of both the radicals and the new immigrants (two distinct but definitely overlapping groups). They were paranoid about a fifth column of Reds or anarchists whose loyalties were thought to be to Moscow, or to other countries-anything and everything, but to the United States.

Interesting historical tidbit: the Communist Party USA was one of the first political organizations in the US to actively support civil rights for black Americans (in the Jim Crow South as well as the industrial North).