Okay, I'm going to clarify some things here. For reference, I am a Marxist, particularly a Marxist-Leninist. I used to consder myself more of an Anarchist, but reading more Marxist theory and history books generally led me towards Marxism-Leninism. As such, the explanations I am going to give in a second are from that perspective, a Marxist that at one point considered themselves to be an Anarchist.
All Communists are, first and foremost, Socialists. Socialism is categorized by an economic system where public ownership and planning is primary and thus dominant over markets. Communism refers to a post-Socialist economic system where all property has been collectivized in a world Socialist republic, the famous "Stateless, Classless, Moneyless Society." When I reference the ideology of AES states, I reference Marxism or Communism or a specific strain of Marxism, but when I reference the economic model of an AES state, it changes.
For example, the PRC is Marxist-Leninist, but practices "Socialism with Chinese Characteristics," which appears as a form of a "Socialist Market Economy." This economic model looks different from, say, Cuba, even though both are guided by Marxism-Leninism and working towards Communism.
There are other forms of Socialism, however in the grand historical and theoretical context the overwhelming majority fall into the broad categories of Marxism and Anarchism.
Does this all make sense so far? If you're interested, I wrote an introductory Marxist reading list, the first section in particular is short and very helpful for just being familiar with general terminology and goals.