These two words are often spoken by
the news media and we hear them on some TV shows, but they are never
explained.
The Deepweb refers to part of the Internet, specifically the world
wide web (anything that starts www) that isn't indexed by search
engines, and can't be accessed by Google.
The Darknet refers to non-www networks, where users may need
separate software to access them. For example, Silk Road and many
illicit markets are hosted on Darknet networks like I2P and Tor.