There’s a lot of confusion about what these terms mean. This is how I’ve always separated them, but others’ opinions may (and do!) differ.
Marketing is the paid-for stuff: the catalogues, magazine ads and trade ads which position a product in the market with the aim of telling trade buyers and end-users that the product exists and what it is.
Publicity encourages debate: it gets a product (in this case, a book or its author) written about in articles, discussed or interviewed on TV or radio, or otherwise talked about through user-groups like book clubs.
PR (public relations) centres around improving and enhancing the target group’s perception of a person, a business or a corporation. The target group whose perception is being enhanced is sometimes the general public but it will more often consist of a far more specific group such as a company’s shareholders before an AGM, or parents when a new baby milk or disposable nappy is launched.
Promotion usually involves more legwork than the other three: for a writer, it involves book signings, giving interviews, and giving talks to interested groups. It tends to be focused on smaller groups (unless you’re JK Rowling).
These categories all depend on each other to work their best: you can’t hope to maximise your sales if you neglect any one area. An author can’t carry out a book signing (promotion) without the bookshop knowing about the book (marketing), knowing that the author is available for a signing (publicity), and caring enough about her to consider inviting her in (PR).