As you might be able to tell from http://twitter.mailana.com/ I'm a big fan of displaying relationships between people as a graph. Here's a few of my favorite examples:
http://www.neuroproductions.be/twitter_friends_network_browser/ has a very compact tool for browsing through your friends and followers. It shows the latest message from each person, and it's very quick and simple to browse through the graph. Unlike Mailana it doesn't use your conversations to figure out the graph, just the list of people you follow.
Muckety is a fascinating experiment that uses its database of connections between people and institutions to illustrate news stories. You can manipulate the graphs and use them to explore deeper links. I haven't found a news story that's a killer app for this (Oil Change USA have something similar with very targeted graphs showing how oil money interacts with politics), but I would love to see more journalists using tools like this to show rather than telling.
The TouchGraph folks have been doing amazing work for years with their Java visualization applet. My favorite demo is probably their Facebook graph, but check out their Google tool if you want to see the connections between websites.
For a slightly mysterious Facebook network visualizer give Nexus a test-run. It shows a lot of detail on your friends and their connections, and I'm very fond of the straightforward interface.