What is a post-print of an article or research paper, and what is it used for?
A postprint is the final version that is given to the journal for copy editing and typesetting.  It includes changes made in the refereeing process, but not the journal's typesetting.  It is often referred to also by the phrase "author's final version".

In contrast, a preprint most specifically refers to a manuscript as it was before peer review.  However, the term preprint is often used more generally (for instance, when referring to a preprint server) to refer to any version of the manuscript besides the journal's final typeset copy.  Thus it is common to find postprints on arxiv.org, which is commonly referred to as a preprint server.

Postprints are mainly used as a way to provide green open access. Many publishers allow authors to distribute postprints through their own website, an institutional website, or a preprint server.  Many institutions (e.g. Harvard) assert a non-exclusive right to distribute postprints written by their employees, regardless of publishing agreements.