In broad terms, a publicist is someone who will help you sell your story to media channels and get the word out about your book. This can mean contacting news agencies, radio and television channels, and spreading the word on social media—blogs and podcasts, the Twittersphere, LinkedIn, etc.
In fact, story and genre considered, publicity can mean getting your story to all forms of media, traditional or otherwise. Your publicist might get your story out to magazines, blogs, vlogs, or even talk shows.
Your story can mean anything from your book to your personal story (i.e., the hows and whys of writing the book), or even both, depending on how novel or marketable that story is.