No it sounds like you’re pretty right on. When no one owns anything, it doesn’t create economic divisions in society.
As for the other part of your question, the wiki says it better then I can:
In the first manuscript, Marx exposes his theory of alienation, which he adapted from Feuerbach’s The Essence of Christianity (1841). He explains how, under capitalism, more and more people rely on “labour” to live. That is, before people could rely in part on Nature itself for its “natural needs”; in modern society, if one wants to eat, one must work: it is only through money that one may survive. Thus, if the alienation of the worker consists in being a “slave toward its object”, the worker is doubly alienated: “first, he receives an object of labour, that is he finds work [as one says: ‘I finally found work!’], and second, he receives means of subsistence. He thereby owes it [to labour] the possibility to exist first as a worker, second as a physical subject. The last straw of this servitude [or serfdom] is that it is only his quality as a worker that permits him to continue to conserve himself as a physical subject, and it is only as a physical subject that he can be a worker”. In other words, the worker relies on labour to find money to be able to live; but he doesn’t simply live, he actually only survives, as a worker. Labour is only used to create more wealth, instead of achieving the fulfillment of “human nature”. This intervention of the concept of “human nature” has also been one of the long-standing factors in this text’s being largely ignored, as it seemed too “humanist” and therefore akin to liberalism and bourgeois philosophy (in a literal sense: a philosophy founded on the bourgeois rights of Man proclaimed in the 1789 Declaration of the Rights of Man and of the Citizen).