Basic chart and graph applications
Datawrapper Upload your data (as CSV) or copy/paste it into this free web application to make a quick chart that is publishable to the web.
Google Public Data Easily create line, bar, and bubble graphs from publicly available data sets and metrics.
More sophisticated options
Tableau Public Free downloadable software that works well with survey data, allowing you to filter data and recode variables, as well as create a variety of charts and graphs.
SDA (Survey Documentation Analysis) Developed at UC Berkeley, SDA is a web-based tool that is integrated into various collections of survey data, including the General Social Survey (GSS) and the American National Election Study (ANES), as well as microdata found in the Integrated Public User Microdata Series (IPUMS) which includes a more fine-grained version of Census data.
Gephi Open source platform for visualizing networks and complex systems, including dynamic and hierarchical graphs.
Fully featured analytic software
The following professional-grade software packages allow you to manipulate and visualize quantitative data in very complex ways:
SAS Powerful statistical analysis and data management tool, uses command-line interface, steep learning curve
Stata Powerful statistical analysis, not as strong at data management as SAS, good combination of ease and power
SPSS Relatively sophisticated statistical analysis tool with a user-friendly GUI
R Free, versatile option for data analysis and visualization
How do I choose?
Choosing statistical software (online video, 36 minutes)
Detailed comparison between SAS, STATA, and SPSS (Institute for Digital Education & Research, UCLA)
Themed visualization applications
World Bank Data Visualizer uses "bubble charts" to display social, economic, and health data from over 200 countries.
Gapminder is a tool for exploring and comparing global development indicators
Sunlight Foundation makes available a number of visualization tools related to politics and transparency, including "Lobbying Tracker" which tracks lobbying activity and "Capitol Words" which explores the most popular words and phrases used by legislators in the U.S. Congress.