Communism is a social and political ideology which is a strict form of 
Socialism.  The concepts behind modern Communism and Socialism are best attributed to 
Karl Marx and Friedrich Engels.  In its pure theoretic form, communism exists without a state and the individual is in all ways equal to every other individual.  There is no individual ownership, only community (hence the word communism) ownership.  Ideally, there is no centralized government structure, as that would place certain individuals above others.
EditSpeech Relevance
Reagan never directly references Communism in '
A Time For Choosing' but its influence throughout the speech is there as he references the likes of 
Lenin, 
Marx, 
Stalin, 
Khrushchev and 
Castro.  Meanwhile, in '
Encroaching Control' there are a handful of direct references and its influence is greater.  It would be difficult to quote them without simply copy-and-pasting the entire speeches.
At the time Reagan gave both speeches, the United States was deep in the 
Cold War with the world's only other superpower, the 
Soviet Union.  The 
Soviet Union was the most powerful communist country in existence.
EditSource Links 
Communism (Wikipedia)The Difference between Communism and SocialismCommunism vs SocialismEditSome Available Texts
Additional information about Communism is available in e-books which this Wiki's Administrator has restored and made available to purchase through Amazon.
Revolutionary Communism in the United StatesRed War on the Family