There are actually quite a few applications for visualizing and analyzing graphs:

 - [Gephi][1] and [Cytoscape][2] are two well-known open source
   applications that support large and complex graphs.
   
 - If you're mainly interested in visualizing graphs, have a look at
   [Graphviz][3], which is an absolute classic. For interactive Web based visualizations of graphs created with Gephi you can use the JavaScript library [sigma.js](http://sigmajs.org/).
   
 
 - You can also use [R][5] or commercial tools like [Mathematica][4] if you're 
   more interested in the statistical and analytical aspects (see also [this 
   question over on Stats SE][6]).


  [1]: http://gephi.org/
  [2]: http://cytoscape.org/
  [3]: http://graphviz.org/
  [4]: http://www.wolfram.com/mathematica/new-in-8/graph-and-network-analysis/
  [5]: http://www.r-project.org/
  [6]: http://stats.stackexchange.com/questions/6155/graph-theory-analysis-and-visualization