A builder builds. A dancer dances. A publisher publishes — which is to say, oversees the printing, editing and, most importantly, the selling of books, magazines, websites, and even music.
It's a matter of debate who is more important at a magazine, the editor or the publisher, but I'm sure you could get convincing arguments for either. The editor is in charge of content, while the publisher runs the business side of things and often owns the operation, too. The publisher of a newspaper is often also the owner. When a writer wants to get their book published — printed and sold — they send it to a publishing house, which may become the book's publisher.