The Society for Editors and Proofreaders describes the copy editor's job very well. We do not try to re-write the book, but instead correct errors in grammar, spelling, punctuation, usage, and, depending on the author's needs, style. We will ensure proper flow and make sure that the author isn't omitting anything: the author knows the material so well that she may inadvertently leave out important information. We make decisions about consistency, especially in novels that use dialects. (I edited a Scottish time-travel romance [not Outlander, though I sure wish it had been!], and let the hero use "No" as well as "Nay" throughout the book. I use "No," "Nope," "Not a bit," "Not at all," "No way," and "Nuh-uh," so didn't consider it important that the hero used "No" sometimes and "Nay" other times. I was overridden by the head editor, though.)
Some of what I do as a copy editor can also fall under "line editing" but I do not go into structural editing. I assume that the author knows what he wants to say, and I make it as clean and tidy and internally consistent as possible! If you have any questions about what I can do for you, please use the Contact section to inquire.