Suppose I have the following sentence:
I lived in a communist country during the fall of communism in 1989.
Should "communist" and "communism" be capitalized?
The correct capitalization for that sentence is:
I lived in a communist country during the Fall of Communism in 1989.
(You may want to keep "fall" in lower case.)
A proper noun should be capitalized, as it is a unique entity. A common noun represents a class of entities or non-unique instances of that class, according to the English SE tag for proper nouns. Applying that definition, "a communist country" is an example of a common noun. It should not be capitalized.
The other occurrence of communism in the sentence is a specific instance. (I presume that it refers to the former Soviet Union? Or another nation, doesn't really matter which one). It is unique by virtue of the fact that it was "of 1989". That is why the second occurrence of communism in your sentence should be capitalized.
Communism is a common noun and, as such, neither it, nor any adjective derived from it, normally needs to begin with a capital letter. If, however, the adjective is part of a title, as in The Communist Party of Trashkanistan or The People's Communist Republic of Transculpania, or if the noun appears in a slogan such as Workers of the World Fight for the Victory of Communism, then the initial capital would be in order.
In your example, communist is a description so has no capital letter, just as the USA is a democratic country (and has a republican form of government) whether the current government is Democratic or Republican.
Communism is more complex, and I think it depends slightly on your politics. Small-c communism is a political idea, named (probably) from the Paris Commune of 1870, but going back at least as far as the early Christian Church. Lenin, in 1917, set up a state on avowedly communist (as opposed to democratic or monarchical) principles. This became a multi-national, totalitarian system going by the name of Communism (or, internally, Socialism as in US S R) but certainly not communist. If you think the 1990 fall of that system discredited the idea, it should be a small c; if it had no more to do with communism than the German Democratic Republic had to do with democracy, use a capital.