For US Data- different levels of analysis (country, state, city, health district, census tract, zip code), decide on appropriate level based on research question. Different sources of info are aggregated at different levels.
Population Reference Bureau (PRB) AmeriStat web site has easy to access state information
PRB Introduction to demography: Population: A Lively Introduction
Infoshare: This website has the latest census counts and estimates, aggregated at different levels of geography (state, place/city, census tract, block, zip code).
POPIN, United Nations Population Division (which includes information from various divisions within UN (WHO, UNICEF, UNDP, UNHCR) .
United Nations Statistical Office has national census data- specific to country, You can also try searching on line using the name of the country as part of the address, i.e. Japan