Just like an astronaut navigates the stars (still only figuratively, unfortunately), an internauta navigates the complex network of tubes connected computers collectively known as The Internet. The threshold to be considered an internauta is more or less being able to send and read email and watch a video on YouTube. I mainly hear the term on the television news when talking about users of a certain website. The Spanish wikipedia page for internauta vaguely claims that the term originated in France. The English wikipedia page for internaut claims that its use is spreading, but I haven't ever seen it written or heard it spoken in English. Personally I'm all for its spread.
I've always despised the term "web surfer". The term was borrowed from someone who rapidly changes television channels, and I can only assume that it originated there (or maybe in pre-television radio?), as a clever metaphor for "looking for a good radio wave".