Most international development agencies have databases with a variety of indicators that do not require any statistical analysis and can be used by anyone. These include demographic data from the U.N; health data from the WHO; development data from the World Bank; labour force data from the ILO; and education data from UNESCO.
The Demographic and Health Surveys (DHS) and the Multiple Indicator Cluster Survey (MICS) collect a wealth of demographic and health information and conduct repeated surveys in countries which allows for looking at time trends. These surveys produce detailed reports that present a considerable amount of data.
The DHS also has a STATcompiler - a great tool which allows for online visualization of indicators generated from their surveys. The STATcompiler generates the national indicators but also allows for comparison of sub-groups by characteristics such as sex, place of residence, age group and education.
For people with advanced statistical skills there are many household surveys that can downloaded for free. Survey data allows for analysis at the individual level as opposed to the aggregate data that the databases provide. The World Bank has a catalog of over 3,000 household surveys in their microdata library.