Self-publishing is the publication of any book or other media by the author of the work, without the involvement of an established third-party (traditional) publisher. The author is responsible and in control of the entire process including design (cover/interior), formats, price, distribution, marketing & PR. The authors can do it all themselves or outsource all or part of the process to companies that offer these services.
A self-publishing company (e.g. Aura Publishers) offers the service of self-publishing on behalf of the author at a fee.