Although Karl Marx frequently denounced ideology in general, there is no evidence that he ever actually used the phrase "false consciousness". It appears to have been used—at least in print—only by Friedrich Engels.
Engels used the term "false consciousness" to address the scenario where the ideology of the ruling class is embodied willfully by a subordinate class. "Consciousness", in this context, reflects a class's ability to politically identify and assert its will. The subordinate class is conscious: it plays a major role in society and can assert its will due to being sufficiently unified in ideas and action. Engels dubs this consciousness "false" because the class is asserting itself towards goals that do not benefit it.
It's worth noting that "ideology" in this context does not refer to the typical usage of the term. Ideology, as used originally by Marxist theorists, refers to the cultural beliefs and accepted wisdoms that a society creates and holds as a reflection of its internal dynamics, that is to say the superstructure of the society. The definition has since drifted, such that outside Marxist theoretical discussions "ideology" is nearly synonymous with worldview or personal philosophy. Marxist theory, however, continues to use the original definition.
Marshall I. Pomer has argued that members of the proletariat disregard the true nature of class relations because of their belief in the probability or possibility of upward mobility. Such a belief or something like it is said to be required in economics with its presumption of rational agency; otherwise wage laborers would not be the conscious supporters of social relations antithetical to their own interests, violating that presumption.