U.S. Census Bureau International Data Base This site has "statistical tables of demographic, and socio-economic data for 227 countries and areas of the world."
Eurostat A wide range of statistical information from the European Union and its member nations.
OFFSTATS: Official Statistics on the Web Links to statistics from all over the web arranged by country, by region, and by topic.
UNESCO Institute for Statistics Data for Sustainable Development Goals
U.C. Atlas of Global Inequality Statistics on a wide variety of indicatiors, health, economy, gender, etc. for countries all over the world, generally for 1960 and forward. Usually stats can be displayed and downloaded.
NationMaster Statistics in over 300 categories from almost every country in the world. Comparisons of countries can be done. Also has brief general profile and map of each country. There's a wiki-ness to this database, so use with caution. Verify info found here.
State of the World's Children: UNICEF Each year UNICEF gathers statistics on a topic concerning the world's children.
UNdata Searchable data from the UN.