A linear book is one where events unfold in a strict chronological order. Linear books are easier for dyslexics and other less able readers to read. It was one of the reasons postulated for Harry Potter's success.
Non- linear means the events unfold not in chronological manner -- for example there are extended flashbacks. These sorts of books can be highly compelling but they are difficult to write properly. The ability to seamlessly weave is more difficult than it looks.
I have read and enjoyed both. Because my books are commercial fiction, I try to write linear books. If I used an extended flashback, it would be because I could not figure out how to write it in any other way, AND I would probably write it as a prologue (thus making it linear). But there are times when it is absolutely necessary. ANd it can add a lot, but they are not probably as many as most authors think.
Some people like to think that in order to do a reunion story, you must have extended flashbacks. You do not. You do need to show the effect of past mistakes on present behaviour. And at some point you need make sure the flashback is actually needed by the reader rather than it being necessary for the writer. I have often sketched out what really did happen between my characters in the back story. For example, I know exactly how Sela and Vikar broke up and why and what was said. The reader only gets a few snippets in Viking Warrior Unwilling Wife. In the Viking's Captive Princess, Thyre's mother's story is important and I had to sketch it out. I could actually write a book on it. The reader again only gets the important snippets -- ie those things that drive the present story forward.
Everything has to drive the present conflict forward.
My current wip also is a reunion story without a flashback. I had to sketch out a number of things and the addiction to flashback is tempting BUT I want to make sure the conflict is in the present and not the past. The past is only there to clarify and influence, not dominate. If it dominates, then it should be a linear part of the book as I write commercial fiction that is easy to read.
I have a post up at the Pink Heart Society about voice and my current obsession with Hallelujah.