Communism is a very broad term that encompasses lots of different types of people: Marxists obviously, but also many (if not most) anarchists and many futurists (like yours truly) as well.
I don't know much about the various schools of anarchist communism, admittedly. The only big-name anarchist communists I'm familiar with the material of include Alexander Berkman and Noam Chomsky. I think the relatively popular '90s rap-rock band Rage Against the Machine was also in the general anarcho-communist train of thought kinda sorta, though none too intellectually particular about it.
As an ex-Marxist, I'm intimately familiar with many varieties of Marxism, though it can be observed that the schools of Marxism all fall into one of four broad super-schools, which include, in order of invention, orthodox Marxism, Leninist Marxism, critical theory, and analytical Marxism. Most Marxists are very socially liberal, to answer one question from the OP. The exceptions to that rule most often come out of the orthodox and Leninist traditions, though there are many hardcore liberals among their ranks as well.
There are also futurist communists (like me). The most prominent futurist communist organization out there is Peter Joseph's
Zeitgeist Movement, which has something like 500,000 members worldwide. As for me, I kind of fuse some elements of Marxist theories on historical progress in with Peter Joseph's thinking on mechanization having the potential to create a jobless, moneyless, stateless, communist future. Both the average Marxist and the Peter Joseph crowd broadly consider themselves scientific socialists (not humanist ones) and so do I.
There are also primitivist communists who are the opposite of the futurist ones. They aspire to return humanity to a tribal existence. They're not very popular.
As to what communism is in the abstract, it's a situation wherein all property is owned in common.