communism is an economic system whereas democracy is a social one

Communism is a political and economic ideology whose goal is the creation of a communist society, the pseudoscientifically postulated utopia of a stateless, classless, moneyless, post-scarcity society. Communist ideology is like the Christianity of politics & economics that keeps promising the 2nd coming of Christ: they insist it’ll happen someday inevitably. No possible way Marx was wrong.

Colloquially, communism refers to a communist state (also known as a Marxist–Leninist state): a political system/government consisting of a socialist state following Marxist–Leninist political philosophy with a dictatorial ruling class that promises to achieve a communist society.

Democracy is a political system/government in which political power is vested in the people or the population of a state. Colloquially, democracy refers to liberal democracy, also called Western-style democracy, or substantive democracy: democracy following ideas of liberal political philosophy.

So, colloquially, communism refers to a political & economic system whereas democracy refers to a political system.

As a political system, the communist state is totalitarian, the most extreme authoritarianism:

Totalitarianism is a label used by various political scientists to characterize the most tyrannical strain of authoritarian systems; in which the ruling elite, often subservient to a dictator, exert near-total control of the social, political, economic, cultural and religious aspects of society in the territories under its governance.

Whereas an authoritarian regime is primarily concerned with political power rather than changing the world & human nature (they will grant society a certain degree of liberty as long as that power is uncontested), totalitarianism aims for more. A totalitarian government is more concerned with changing the world & human nature to fulfill an ideology: it seeks to completely control the thoughts & actions of its citizens through such tactics as

• Political repression: according to their ideology, rights aren’t inherent or fundamental, the state is the source of human rights. Rights (eg, freedom of speech, assembly, & movement) are suppressed. Dissent is punished. Unauthorized political activities aren’t tolerated.
• State terrorism: secret police, purges, mass executions & surveillance, persecution of dissidents, labor camps.
• Control of information: full control over mass communication media & the education system to promote the ideology.
• Economic control.

All of this is entirely compatible with Marxist-Leninism.

Liberalism, however, is fundamentally incompatible with authoritarianism. It holds that governments exist for the people & authority is legitimate only when it protects inalienable/fundamental/inherent rights & liberties of individuals. The people have an inherent right to obtain a government with legitimate authority, and when their government lacks or loses legitimacy, the people have a right & duty replace or change that government until it obtains legitimacy.