Today I ran across the wording "post-print version" of an article (on ResearchGate). I'm aware of terms pre-print, in preparation, under review, published, etc., but I have never seen the term "post-print" before. What does it mean and what else should we know about it (i.e., when to use it)?
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.
A standard definition comes from SHERPA/RoMEO:
[...] this listing characterises pre-prints as being the version of the paper before peer review and post-prints as being the version of the paper after peer-review, with revisions having been made.
This means that in terms of content, post-prints are the article as published. However, in terms of appearance this might not be the same as the published article, as publishers often reserve for themselves their own arrangement of type-setting and formatting. Typically, this means that the author cannot use the publisher-generated .pdf file, but must make their own .pdf version for submission to a repository.
For green open access purposes, self-archiving the post-print is preferable to self-archiving the pre-print, as it is the post-print that contains the content improved after feedback from reviewers.
Publishers may use different terminology: For example, Elsevier uses "Accepted Manuscript" instead of the word post-print in its sharing policy.