What is this type of circular-link visualization called? Does this type of chart have a name? More importantly, is there any visualization library I can use to produce it?

http://www.nytimes.com/interactive/2007/12/15/us/politics/DEBATE.html
 A: I found that the dependency graph in Flare is also similar to what I want:
http://flare.prefuse.org/apps/dependency_graph
A: It is called a Chord diagram.
Now that you know its name you can research for the tool that best suits you. I dont think it is nice to advertise tools.
A: I would just add:
As you point out, Flare has the dependency graph, which Aleks Jakulin argued was similar but better.  This was based originally on the "Hierarchical Edge Bundles:
Visualization of Adjacency Relations in Hierarchical Data" (Holden 2006).  
I personally prefer to use Protovis to Flare directly, and you can look at Mike Bostock's example of the same graphic.  Here is also an example of an Arc Diagram in Protovis, which is very similar but laid out linearly.
A: For the #Rstats crowd there are two other options.
circlize library (package, vignette):

This package aims to implement circos layout in R.


RCircos library (CRAN):

RCircos package provides a simple and flexible way to generate Circos
  2D track plot images for genomic data visualization.

A: Take a look at Circos:

Circos is a software package for visualizing data and information. It visualizes data in a circular layout — this makes Circos ideal for exploring relationships between objects or positions.

The flowing data blog also had a post on this that you might find interesting:


*

*Visual Representation of Tabular Information – How to Fix the Uncommunicative Table
