One important point that we at Writer Beware try hard to make is that the traditional definition of "vanity publisher"--a publisher you pay to print and bind your book--is outdated. Sure, there are many straightforward vanity publishers that want cash on the barrelhead and make no bones about the fact that you're paying to be published. But there are also many stealth vanity publishers--publishers that try to sanitize their fees by calling them something else or shifting them to some other aspect of the publication process, or that attempt to deceive writers by failing to disclose upfront that they require a cash infusion.
There are a number of stealth vanity publishers on Writer Beware's Thumbs Down Publishers List. New World Media, a.k.a. American Book Press, doesn't require authors to pay for printing and binding--just for editing (cost: $3,000-$3,500). American Book Publishing also doesn't charge for printing and binding--just for "setup" (cost: around $700). Durban House Publishing doesn't charge for any aspect of book production at all--just for marketing (costs reported to Writer Beware range from $15,000 to $25,000). Harbor House and SterlingHouse Publisher also don't charge for book production--but they do require authors to buy their own finished books, with the money due on contract signing (1,000 copies and 550 copies, respectively).
The only "traditional" vanity publishers on the list are Helm Publishing and Tate Publishing. Their fees (Helm charges $750-1,500, and Tate asks for $3,985) are clearly identified as being for printing and production. There's a catch, though: authors don't find out how much they have to pay until after they've submitted.
Other stealth vanity tactics, from Writer Beware's archives: the now-defunct NovelBooks at one point bound ads into its books, requiring authors to sell ads for their own books and pressuring them to buy ads for other authors' books. Picasso Publishing, which closed its doors a couple of years ago, made its authors pay for a publicity campaign. Authors with Gardenia Publishing (also out of business) had to sell a minimum quantity of books prior to publication--they didn't have to buy the books themselves, but if the money wasn't turned in to the publisher, publication was off. Still other stealth vanity publishers only accept submissions through the publishers' own paid assessment services, or keep all royalties until the book's production costs are reimbursed, or force authors to buy "adjunct" services (such as cover art or interior design) from approved vendors.
A special variant of stealth vanity publishing is the "subsidy" or "joint venture" or "cooperative" or "partner" publisher. Such publishers claim to match your fees with their own money, or to contribute goods and services of substantial value. In other words, you aren't paying the whole freight. However, while there are a few genuine subsidy publishers (mainly in specialized fields such as academic publishing), a claim of subsidy or partnership publishing is much more likely to be a marketing ploy designed to make you feel better about handing over cash. In fact, the fee has probably been carefully calculated to ensure the publisher's profit, and is far more than the actual value of the services provided. Subsidy publishers often claim to support their authors with significant marketing efforts, but they have little incentive to market their books, since they've already been paid upfront.
Recently, a new wrinkle in the stealth vanity publisher game came across my desk. Blue Dolphin Publishing presents itself as a "traditional" publisher. No fees are mentioned in its submission guidelines. Authors who submit, however, receive a "Dear Author" letter informing them that they have "a worthy project" but in order for their book to be published there must be "a separate contract with an outside investor." While Blue Dolphin "never expects an author to finance a project," they "do, however, ask the author if he or she knows anyone who can help us."
This is clever psychology. It isn't an actual demand for cash, nor is the author herself being directly asked to "invest." Heaven forbid! That would be vanity publishing! But the ball is now firmly in the author's court, and the publisher is counting on the carrot of publication to inspire her to "help"--either by raiding her own bank account, or borrowing money from a relative. (Complaints Writer Beware has gotten about Blue Dolphin suggest that this is exactly what some authors do.) So the publisher never actually requests money--but obtains it even so. Stealth vanity publishing indeed.
Bottom line with the stealth tactics described above: you're paying to be published. Don't fall for deceptive terminology, and don't be fooled by elaborate rationales. A publisher that requires you to lay out your own cash for ANYTHING, at ANY point in the publication process, is a vanity publisher. Period.
For a detailed discussion of why vanity publishing is never a good idea for writers, see the Vanity Publishers page of Writer Beware.