Perhaps before any contemporary open access movement, there was arXiv, a moderated pre-print archive for Physics, Mathematics, Computer Science, Quantitative Biology, Quantitative Finance and Statistics, meant for perpetual access. It started in 1991 at Los Alamos National Laboratory (USA). Now it is hosted by Cornell University with annual voluntary contributions.
In contrast, life sciences has had a later start. PlosOne started in 2006. It is significant in that conceptually, it challenges the impact factor based measure of research relevance and importance - an unfortunate conceptual mirror of the web-hits. Instead,
"Each submission will be assessed by a member of the PLOS ONE Editorial Board before publication. This pre-publication peer review will concentrate on technical rather than subjective concerns and may involve discussion with other members of the Editorial Board and/or the solicitation of formal reports from independent referees. If published, papers will be made available for community-based open peer review involving online annotation, discussion, and rating"
The first Open Access Week was held in 2008.
2011 highlighted alternative efforts towards making academic papers available. In 2011, Aaron Swartz is arrested for downloading millions of publications from JSTOR by hacking into the MIT network - he committed suicide in 2013 allegedly due to depression stemming from the arrest and lawsuits that followed. The same year in the fall of 2011, Sci-Hub (wiki), the pirate academic publication server started.
Since 2012, EU grants have started mandating that publications borne out of EU funding must be open access.
In late 2013, Cold Spring Laboratories launched BioRxiv, the equivalent of arXiv in biological sciences (excluding medicine) where preprints can be published, to be accessed by all. "For scientists who might worry that posting a preprint will jeopardize its chances at a journal, Inglis points out that one-fourth of bioRxiv’s papers have later appeared in scores of journals, including the most selective."
2013 The Guardian publishes Open Access: 6 Myths to put to rest
Recently, academics have started to take a stand on the publication practices of Elsevir, one of five publishing houses that control more than half of the scientific publications. Funding agencies are following these demands, and starting 2017, Gates Foundation funded projects will be Open immediately, not following the model of the embargo (having to wait several months before articles become open). In 2017 - Major German Universities do not renew Elsevier subscriptions.
There are new open access publication initiatives, such as eLife. Funded by Howard Hughes Medical Institute, Max Planck Society, and the Wellcome Trust. Its economics have been studied and reported, as in the Wellcome Trust's Economic Analysis of Scientific Research Publishing (pdf, from January 2003), and analysis of financial models of open access publishing in life sciences by L. van Dorp (pdf, thesis from Imperial College of London, September 2012). This article from March 2013 discusses the costs of going Open Access. Many other initiatives are now present, including F1000Research.
"The first aim of the initiative is to launch an open-access journal covering the most outstanding advancements – from basic biological research through to applied, translational, and clinical studies."
Movement toward transparency and open access and open data is a very dynamic area. The Open Knowledge Foundation pushes towards open data.
For a more extensive Open Access History, go to this wiki