There are 2 main routes to achieving open access, distinguished by which version of an article is made available: the author's accepted manuscript or the publisher's final version. These are called the green road and the gold road, respectively.
Publishing in an open access journal is known as the gold road.
Finding an Open Access Journal
The Directory of Open Access Journals (DOAJ) is a vetted list of journals that meet specific inclusion criteria.
The Open Access Scholarly Publishers Association (OASPA) requires its members to follow a strict code of conduct.
Article Processing Charges
Some OA publishers require authors to pay a fee for publication, usually to help offset management and hosting costs. PLOS is a good example of one such publisher. However, only about 30% of open access publishers currently charge a fee, and some funding agencies are willing to pay APCs for any publications they support (check with your funder for details).
Hybrid journals, which make some but not all of their published material open access, are a growing category. Typically, these publishers recoup some of their costs via an APC, in exchange for providing open access to an article in a journal which is otherwise not. Many major publishers now include hybrid options in their portfolio.
The growth of open access and the increasing number of researchers worldwide looking for a home for their manuscripts have empowered some bad actors to turn a profit by enticing researchers with the promise of publication, collecting an APC, and then failing to follow through on their promises. These predatory publishers might never actually publish the work, or they might publish every author who produces the APC, with little to no review or editorial process.
Some individuals and organizations publish "blacklists" of potentially predatory journals; but the safest way to avoid them is to deal with a reputable source listed in the DOAJ or OASPA and look for positive or negative quality indicators. Also, remember that predatory behavior is not limited to open access publishers.
Self-archiving, that is, depositing a copy of your author's accepted manuscript in an open access repository, is known as the green road.
Benefits of Self-Archiving
Taking the green road means you're not limited in where you can publish. Over 75% of traditional publishers allow authors to make some version of their articles available in a repository, though specifics vary. The SHERPA/RoMEO database of publisher policies can help you find out what your publisher allows.
Open Access Repositories
Repositories typically fall into one of two categories: institutional or disciplinary.
Open Works is the College of Wooster's institutional repository. Any affiliated author may post an article.
A huge number of disciplinary repositories have proliferated across the internet in recent years. Major examples include:
Databases of open access repositories, such as OpenDOAR and ROAR may be helpful if you are looking for an appropriate repository for your work. You can also contact a librarian. And remember, Open Works is always an option as well.
Academic Social Networks
Sites like ResearchGate and Academia.edu, which have jump started the category of "academic social networks," allow users to post their papers and have high visibility on search engines. These sites are superficially similar to open access repositories in some ways, but they lack key features that make repositories truly open, long-term solutions for archiving scholarship, as this chart from the University of California summarizes: