Post means after. However the problem with the word post is that it is difficult to know whether it is evoked to signal a stark break with what went on before (post war, postmodernism perhaps) or a development from within a tradition (post-structuralism as a follow on from structuralism). If we want to see post-positivism as a development within positivism, then we might draw on the work of Popper. For here there was an acceptance, even a strengthening, of the case for deductive logic, familiar to positivism, but a new willingness to concede that what we know is fallible and generated within a particular context.
In more recent times critical realism represents another development of positivism. Critical realism has been described as post-positivist as it accepts the ontological idea from positivism that there is an objective reality but introduces an epistemological relativism that undermines the positivist project. You can take your pick in arguing how far critical realism has strayed from positivism.
Some see post positivism as offering a reconciliation of interpretivism and positivism. In Henderson (2011) post-positivism could equally be seen as post interpretivism for she sees it as addressing overly subjectivist accounts of ontology and epistemology that appear in interpretivism. In fact Henderson makes the important point that while many social researchers use the term interpretivist to describe their work they are not really interpretivists as they implicitly hold on to quite conventional ideas about ontology and what constitutes methodological rigour. There is an overlap in her and other accounts of post-positivism (e.g. Ryan, 2006) with the recognition of claims of reflexivity, flexibility, concern for audience and commitment to mixed methods. Finally many of the key concerns of post positivism were rehearsed in Pragmatism though pragmatism offered a distinctive commitment to action oriented research.
Henderson, K. (2011) Post-positivism and the pragmatics of leisure research, Leisure Sciences, 33:4, 341-346.
Ryan, A. B. (2006). Post-positivist approaches to research. In M. Antonesa, H. Fallon, A. B. Ryan, A. Ryan, & T. Walsh, with L. Borys, Researching and writing your thesis: A guide for postgraduate students (pp. 12–28). Maynooth, Ireland: MACE, National University of Ireland. [Online] http://eprints.nuim.ie/archive/00000874/