Visualization of graph data is incredibly challenging, particularly when it comes to extremely large, scale-free graphs and social networks. A few simple searches on the Web and you will find some mesmerizing and very cool images. Perhaps the most cited and fascinating of these are Matt's "Mapping the Blogosphere" series.
Some of my favorite tools for graph visualization are LGL (see Twitter visualization using LGL), Prefuse, Guess and UCINet (no specific ranked order). I have also been waiting to get a chance to try out AT&T's large network visualization software. Two other promising tools that I have not played with yet are Tulip (TOTH Eytan!) and yWorks. Following is an image I had generated a year ago (more here) using UCINet. This is a community of political blogs discovered using an implementation of Dr. Mark Newman's community detection algorithm:
One thing that I think would be exciting is visualizing evolving networks. Current tools force you to map blogs, social networks etc. into static graphs. This is a serious limitation. Their real meaning and value comes from seeing the temporal interactions between members, evolution of community structures, changes in strength of ties over a period and detecting bursts. As far as my knowledge goes, this feature is not supported by any of the open source/freely available tools that I have explored. I would love to hear about other graph visualization tools that I may be unaware of.