...Government control or ownership of the factors of production (depending on who you ask). What that means, well, ask a 100 economists and you will get at least 300 different answers. Karl Marx never actually gave any blueprint for Socialism and even less so for Communism. It was philosophical, people forget or don't know he was also a philosopher.
There is the Nordic Model, but as the old joke goes:
Person A: Socialism doesn't work
Person B: The Nordic nations are socialist and are doing fine.
Person A: They are not socialist, they are capitalist with strong social welfare programs.
Person B: Then lets adopt those programs.
Person A: No! That's Socialism!
China and Vietnam have walked away from Communism and have embraced the theory of Market Socialism. And a couple of countries in Europe are starting to slowly adopt it as well. Ultimately, it works and that's what the elite wealthy class fears.
We treat economic systems as some sort of religious dogma, when in actuality, it's an evolution. Society and its values evolve, thus economic systems must evolve with them.
For the Billionaire class, they just accumulate wealth for the sake of accumulating wealth. And they are afraid of change that will stop them in that endeavor. The idea of "I can get rich" no longer resonates as many in lower classes have realized, its impossible to get that rich no matter what they do. So now the focus is on a better and equitable society. But that means the Billionaire class will have to lose a chunk of its wealth and they don't want that.
There is this always prevailing myth that there is this infinite amount money, when in reality, it is finite.
As to why the wealthy fear it, ultimately, its a change and no one likes a change in the rules.