What is Communism? And why are so many Americans afraid of Communism and Socialism?

Communism is an economic system originally invented by a German named Karl Marx in the middle of the 19th century, based on ideas that had been floating around Germany for the last hundred years or so. Remember, the mid- to late-19th century was a time of very great income inequality; the "robber baron" period in the United States, when men like Vanderbilt and Carnegie and JP Morgan built huge fortunes on the backs of large masses of workers. Today our billionaires are mostly the beneficiaries of technology; Bill Gates, for example, primarily benefitted from inventions that are fundamentally of the mind. The rich men of the 19th century, conversely, built their fortunes mostly on resource extraction and heavy industry (coal, steel, railroads). That meant there was a huge workforce making what we could consider to be nearly slave wages today.
Marx's basic idea was that unfettered capitalism would continuously result in the rich getting richer. And, until certain philanthropically-minded men (most notably Henry Ford) decided to basically govern themselves, Marx was right. His political philosophy, "communism" was based on resource redistribution through a planned economy. Instead of letting people do whatever they felt like and letting the "cream rise to the top," in a Marxist society people would be assigned tasks that best suited their skillset. In theory this would better capture the economic potential of all humanity and the rising tide would lift all boats.
In 1918 the Bolsheviks (a working class political party/terrorist group) in Russia overthrew their King (Czar Nicholas II) and put Marx's theories to the test. They were led initially by Vladimir Lenin and Leon Trotsky, as well as several others. Over time, Lenin accumulated what amounted to absolute power, assisted by other powerful revolutionaries who he appointed to the Politburo, which was the governing body responsible for essentially allocating all resources in the new, planned economy of the Soviet Union. A "soviet" was originally meant to be a small council of local officials, kind of like a city council, that would allocate resources more efficiently at the local level. Thus in theory the Soviet Union was just a large mass of small councils each acting both independently and in concert to help humanity achieve its potential by maximizing the best use of economic resources, kind of like an ant hive.
The experiment did not go well. While the planned economy is a great idea on paper, humans do not act like perfectly rational automatons. As men at the top accumulated power through their control over the economy, they grew corrupt. Dissenters--men with ideas different than those on top--were purged; either killed outright or sent to work camps in Siberia. Leon Trotsky wound up fleeing to Mexico after he fell out with the Soviet leadership; he was assassinated there.
When Lenin eventually died, power fell into the hands of Josef Stalin, who was a classic dictator wielding the doctrine of Marxist-Leninism as a mere facade, to cover up his murderous campaigns. Stalin killed more "undesirables" (mainly an ethnic group called Cossacks) in the Soviet Union than Hitler did Jews. And Stalin's economic planning was poor; he set a series of "5 Year Plans" throughout the 1930s and 1940s which generally failed. Millions starved to death. Millions more died fighting World War II, which ironically propped up the Soviet system and alleviated some of the economic stress by reducing the numbers of mouths to feed after the war.
People hate Communism today because it is a pernicious philosophy. It is founded on a fundamental belief in the goodness of man, and on very respectable rational ideas about the best use of our human resources. But like all Utopian philosophies, it fails to account for human nature. While it has some very good ideas, Communism historically has ended up in dictatorships in every single country where it was implemented. (Cuba, North Korea, China, Romania, Czechoslovakia, East Germany, Poland, and the USSR itself).
Socialism meanwhile is essentially a less extreme version of communist philosophy; a hybrid, if you will, of pure capitalism (where the market dictates people's action and the "cream rises to the top" while people on the bottom sweep streets), and of communism. Socialism or some variant thereof is the predominant government system in Europe today, and arguably even in the United States. The "social safety net" programs like Social Security and Medicaid, along with the graduated income tax (where the rich pay more because they can afford to contribute a greater proportion of their income to help run our government) are socialist ideas and programs.
Finally, most of the pejorative words like "pinko" or "socialist" or "communist" that you hear tossed around by people like Rush Limbaugh are just that: pejoratives. Name-calling. Limbaugh certainly knows all of this stuff. He is a very smart man. His audience, on the other hand, is going to range from people who know it cold down to people who have no idea but just know it's a bad thing to be.