According to Wikipedia, "self-archiving, also known as green open access, refers to the practice of depositing articles in an open access repository, this can be an institutional or a disciplinary repository such as arXiv." Typically, researchers archive pre- or post-print versions of a journal article so that those without a subscription to the journal can still have access to the article. Not all publishers allow this, but the practice is increasingly common and is a good way to foster accessibility even if you choose not to publish in an OA journal.
Photo credit: Delde Open Access Journals
Many publishers now allow some form of archiving locally (see SHERPA/RoMEO). This means that authors can deposit their work voluntarily in repositories at their own institutions or funding agencies, or a disciplinary repository. Sometimes the publisher's version of the PDF may be included in a repository, but commonly authors need to contribute their final, peer-reviewed version of the article.
Iowa Research Online, UI's institutional repository, was created to organize, preserve, and increase the impact of scholarly and creative work at The University of Iowa. It is a dynamic archive of the research produced by faculty, researchers, and students, from published articles in peer-reviewed journals to presentations, theses, dissertations, and unpublished papers. Many institutions all over the world have repositories.
Image credit: Repository66.
Green Open Access encourages researchers to archive their work, either in their institution's repository or in a repository specific to a certain academic discipline. Examples of disciplinary repositories include: