There are a few ways to visualise internet and the traffic on it.

One way is to map it. The BBC has a good article on this.

The following programmes take a starting point and use linking for distance. The amount of links to and from a place is used for size.

I/O/D 4 (the webstalker) is one of my all time favorites, but lacks 3D representation, scaling and movement.

Surf 3D has several different viewing modes, and allows you to move through the map. Unfortunately the interface is a bit cluncky, allthough it gives you loads of options.

Here is a large list of links to mapping projects

The above can lead to complex maps, usually starting from a single point (a URL).
More simply, smaller datasets can be mapped.

3D Traceroute turns a traceroute into a complex grapical representation.

Xtraceroute shows a 3D globe and shows the path of the packets over the globe.

There’s a visual whois programme which alsow plots queries over a 3D globe and shows the information next to it.

Another method is to browse traffic and see what it’s doing.

Swarm shows what other swarmers are doing, with the page with the most traffic in the centre.

Driftnet displays the graphics being surfed on websites (and there is an enhancement which also displays mpeg and audio streams) over the network (install on the router)

Webcollage randomly pulls images from internet and throws them up on the screen.

Then there are more freeform methods of viewing the internet.

Driftnet ver.0 (not to be confused with the driftnet above) starts with a URL and displays in hexadecimal and screenshots, using waves to decide where to go next.

A bit late, but apparently due to some patent issues, MS launched an update on April 11 which breaks Flash object interactivity in IE: you have to click the object before it will do your lovely mouseover type stuff. Well done Eolas, you’ve broken a large segment of Internet. This is yet another example of the ridiculousness of the US patent system and another reason to fight it here in Europe.
The link contains a discussion on the subject and fixes for developers.