Basic information on almost every country is available in the Statesman's Yearbook (in print or online) and World Development Indicators (World Bank). On the Internet, you can get for free the:
The CIA's World Factbook;
World Bank's Countries & Regions pages (www.worldbank.org/);
Economist magazine's Country Briefings (www.economist.com/countries);
Library of Congress' Country Studies;
Library of Congress' Global Legal Monitor, which reports recent legal developments;
Demography and Population Studies page from USA People Search;
BBC Country Profiles;
The Social Indicators data posted by the United Nations' Statistics Division;
U.S. Commercial Service's Market Research Library;
U.S. Census Bureau's International Statistics page.
Amnesty International's country reports;
Links to the statistical agencies of foreign counties.
If you're willing to pay, you can get a D&B Business Information Report detailing the creditworthiness of a country, an Economist Intelligence Unit (EIU) report detailing the political and economic situation inside a country, and/or an analyst report on the economic prospects of a country (see "Analyst Reports"). The Alacra Store sells an array of company profiles, economic data, credit and investment research and market research.
For more on specific countries, search for an entry on the individual country.