RawGraphs: Easy to use, drop in data and visualize it in lots of ways.
Gephi: Gephi is one of the three standard data visualization tools for mapping networks. The tutorial using data about characters from Les Miserables is here. And a tutorial for making Gephi spreadsheets is here.
NodeXL: NodeXL is specifically designed for working with Excel files to visualize networks, and it also has built in connections for working with some social media platforms.
Cytoscape: Similar to Gephi but designed specifically with biology in mind. This means it has certain features that Gephi does not, and it is in general a more developed platform.
Processing: Open source drawing and development environment, interesting for teaching as it was originally developed as a tool for teaching programing to visual artists.
Social Network Visualizer (SocNetV): Social network analysis tool, recently updated in late 2016.
yEd: Useful for diagrams and flow charts.
Digital Methods Tools: Lots of different tools for small-scale, quickie data viz here. From the European Digital Methods Initiative (DMI).
Prezi: Although often used in a linear fashion in a manner similar to the hypercard format of Powerpoint, Prezi is a presentation software designed around moving along edges between nodes.
UCINET: An older network analysis tool that is still updated and has an important user community and data sets.
Pajek:Another older network analysis tool from the late 1990's, which brought important ideas into the field.
Manuel Lima's Visual Complexity: Lots of examples here, based on Lima's Book of Trees: Visualizing Branches of Knowledge (2013) and Visual Complexity: Mapping Patterns of Information (2011).
Albert-László Barabási's Network Science (2015). A textbook companion with explanations, examples and tools.
Star Trek Viz: An example of a social network interface based on the fictional Star Trek series and films.
Music Map: An interactive tree graph of the relationships between music genres.
Kindred Britain: Stanford project showing kinship connections between people in British history.
Black Shoals Project: A 'planetarium' that turns companies, their stock prices and relationships into a starry sky visualization.
Edward Tufte: The doyen of data visualization's website. He is famous for his trilogy of books including "Envisioning Information."
"Fifteen Theses on Contemporary Art" by Alain Badiou: Begins with a discussion of Mark Lombardi's work on visualizing covert networks and a high quality image of a network he drew multiple times between 1979 and 1999.
Hans Haacke: 1971 artwork "Shapolsky et al. Manhattan Real Estate Holdings, a Real-Time Social System, as of May 1, 1971" showing Manhattan real estate in relation to developers as a system.
Ed Ruscha, "Every Building on the Sunset Strip" (1966): An example of archival art, attempting to capture the street at a moment in time.
Charles and Ray Eames, "Powers of Ten" (1977): A short film using cascading images to show the relative size of the universe.
Updated Batchelor 1-2017