The surface web is only about 4% of the internet and includes search engines indexed information you can stumble across while browsing the net.
The deep web is like the ocean the iceberg is floating in. This is information that cannot be indexed by search engines, it contains sites that have to be logged into in order to see the information like bank accounts, databases, and archived material.
The dark web is a corner of the internet which is intentionally hidden from casual users of the world wide web. It's only discoverable through the use of certain internet browsers. While it is home to a host of illegal and vile material and activities, it is also used by those who need to "hide" for legitimate reasons.
Ways of the Web: from the University of Illinois - a really great site on understanding how Google and many other search engines work, what a filter bubble is and how to escape it, the deep web, and evaluating web sources!