The surface is what we think of as "the internet": the indexed, publicly available portions of the web that you can Google.
The first layer of the "deep web" is those pages on "the internet" that are for a variety of reasons unlisted or restricted-access, rendering them inaccessible to most: members-only, password-protected content, pages that haven't yet been indexed, new pages, and pages that aren't indexed because of DMCA complaints , pages that contains info about
The second layer consists of the sleazy, clandestine, contraband-type stuff that most people think of when they think of the "deep web", and it trafficks itself through anonymizing software (Tor, e.g.), .onion links, and p2p/p2m networks. This is what most people think of when they think of the "Deep web". It is difficult to access, is creepy, and is therefore notorious. It consists primarily of pirated software and movies, drugs, guns, gambling, offers of contract murder, and illicit pornography. Avoid it. As a wise man once said, "nothing good happens in that part of town after midnight".
The third and fourth layers are where you see the bulk of the traffic, and is where the truly interesting stuff lives. Most information on these layers of the "deep web" is hidden primarily because it is proprietary or classified. The bulk of deep traffic is on alternate and private networks (that is to say, corporate and government traffic). Layer three consists of classified information trafficked on alternative networks such as JWICS, SIPRNet, and NSANet. The fourth layer is internal corporate traffic hosted on private PANs, WANs and LANs, inaccessible unless you're already inside or that network is otherwise connected to the "internet" (the indexed, "googleable" surface web that we use every day).