The term "vanity publisher" was actually coined by the publishing industry way back at the beginning of the 20th century.  It was meant to discourage competition.  Back then, publishers who could use an author's money to print books (an expensive process) could take significant business away from the publishing companies then in business.  By suggesting that such publishers were unscrupulous and that the writers were egomaniacs, the existing industry prevented serious losses.

Today, there are many different forms of self-publishing, but the concept of the "vanity publisher" has stuck.  In general, it has come to mean the type of publisher (sometimes called a book producer) who prints and binds a book in fairly large quantities at the author's sole expense. Costs include the publisher's profit and overhead, so vanity publishing is usually a good deal more expensive than other forms of self-publishing. The completed books are the property of the author, and the author retains all proceeds from sales. Vanity publishers do not screen for quality--they publish anyone who can pay--and they may offer additional services such as editing, warehousing, and book fulfillment. They rarely offer distribution of any kind.

A subsidy publisher (a.k.a. a joint venture publisher, a co-op publisher, a partner publisher, and many others) also takes payment from the author to print and bind a book, but may itself contribute a portion of the cost, as well as adjunct services such as editing, distribution, warehousing, and some degree of marketing. There may be some limited screening of submissions to rule out pornography or hate literature, but as with vanity publishers it's not the quality of the author's work that counts. As with traditional publishers, the completed books are the property of the publisher, and remain in the publisher's possession until sold. Income to the writer comes in the form of a royalty.
 

POD publishing has more to do with new technology than it has to do with old ways of self-publishing. Self-publishers who go this route have to spend their own money to earn percentages and royalties based on sales. Because of the lower cost of POD publishing when compared to any other type, many writers find this method most attractive. Most POD publishers do not screen manuscripts and will publish anything. Some recommend editors, and a very few insist on editing. Authors can buy as many or as few of their books as they want or can sell, so there are no garages filled with books to contend with. Because there is less waste, POD is a useful way to get a book into the market at a relatively low cost and then see how well it sells.  Most POD publishers offer distribution and often some type of marketing assistance, but marketing is really up to the author.

Most commonly, the term "self-publishing" is used to discuss publishing that is handled completely by the author. Self-publishing requires the author to undertake the entire cost of publication him/herself, and to handle all marketing, distribution, storage, etc. However, because the author can put every aspect of the process out to bid, rather than accepting a pre-set package of services, self-publishing can be more cost-effective than vanity or subsidy publishing (although maybe not as inexpensive as POD publishing). It can also result in a higher rate of return as well as a higher-quality product (if the author knows what he/she is doing). Unlike subsidy publishing, the completed books are the writer's property, and the writer keeps 100% of sales proceeds.