A vanity press, as far as it's been explained to me, is a publisher who doesn't pay you for your work. Instead, you pay them to publish your stuff. They use terms like "no obligation" a lot.

Sure, they'll publish your stuff but you don't get to see it in hardcopy unless you pay for your own copy. Legitimate places will pay you for your work AND give you a copy of the finished product, at least, that's what I've heard from a few people I personally know who have been published by legitimate places.