Think of an artist, drawing a pencil rendering. They'll start with a rough sketch. Once they get the gesture and outline they want, they refine, develop, erase and embellish. They may draw over the initial sketch, they're final rendering may just be drawn right over that sketch. This is linear editing. Every step is a path, even an erasure is a step forward in a line.

Non-linear editing makes a document that uses a reference to whatever it is you're working with. In most any page layout programs, you're linking to the text, the photographs, the illustrations, rather than imbedding them. This is non-linear editing. You're not working on the "asset". You're making notes referencing back to the asset.

Ok, now... remember the episode of Star Trek where the super-powerful almost-godlike alien uber-creature had an image of the Enterprise in a little glass jar, and he could poink the little toy Enterprise and the real Enterprise would be all "The ship is breaking up, Captain!"? (OK. It's fine if you don't want to admit you remember it) ...but that is non-linear editing.

Why do we care? Non-linear editing allows us to do a whole lot of things.
It allows us, first, to work on a great big file without actually having to manage that big file. (Think the Enterprise story, here...) In Final Cut Pro, you're linking to a big file, moving it, sampling it, altering it, but you're only working on a tiny reference to it, not the actual file.

Non-linear editing allows us to work with a bunch of files in different places. A layout may pull source files from all over your hard drive, and still not actually move them into one place.

NLE also gives us the other side of that edge... we can place one source file into several documents, and when we change or update that one source file we are updating all the end-product documents.

Another thing... it allows us to generate versions of the original source easily and in a non-destructive way. You can take a source drawing, change the background, the color of the pencil, the depth of the stroke all concurrently. Impossible in a linear process!

Final Cut Studio, InDesign, Aperture all use non-linear editing. It's one of the many things that leverages the digital workstation to allow you to accomplish almost unheard-of things in a simple, comfortable process.