To find census records for individuals, use Ancestry Library or HeritageQuest. Both include images of the actual returns, and can be searched by name.
The Census Bureau does not release information about individuals to the public until after 72 years, to protect the confidentiality of the information. So, the most recent decennial census for which the individual returns are available is 1950. Up to 1960 the census taker filled out the information for each household. From 1960 onwards, the forms were mailed to households in advance, and then either collected by the census-taker or mailed back to the Census Bureau.
The surviving original census forms were microfilmed in the 1940s. The originals were then given to state archives or destroyed. The scanned images in Ancestry Library and HeritageQuest were created from the microfilms.
[Note: Ancestry Library contains most of the material on ancestry.com, which is available only to individuals by subscription.]
U.S. Census Published Volumes - PDF versions of published U.S. Census publications. 1790-2010. Available via IPUMS.
Selected Historical Decennial Census Population and Housing Counts
Social Explorer - User-friendly interface for interactive maps and census data reports from 1790 to the present.
Historical Statistics of the United States Wide range of historical data.Tables cover specific indicators such as employment, income, saving, poverty, construction, housing, consumer price indexes, and others.
Current Population Reports and Other Population Reports Index
Provides a convenient listing of Current Population Reports back to 1947, based on the Subject Index to Current Population Reports and Other Population Report Series.
Historical Statistics on the Foreign Born Population of the United States: 1850-2000 Bureau of the Census
Tables include country of birth, mother tongue, race, age and sex at the national level. Also provides nativity data (foreign or native born) for metropolitan areas, counties, and 50 large urban areas.
Population of the 100 Largest Cities and Other Urban Places in the United States: 1790-1990 Bureau of the Census
Cities ranked by population size, beginning with 24 urban areas in 1790 up to 100 areas in 1990.
Microdata:
IPUMS USA - University of Minnesota
Harmonized U.S. census microdata with enhanced documentation. Data includes decennial censuses from 1790 to 2010 and American Community Surveys (ACS) from 2000 to the present. Data can be downloaded in compressed format, and recoding permits users to compare various years.
ICPSR
Provides raw data files for manipulation and analysis. Browse from https://census.icpsr.umich.edu/census/search/studies, Census Enumerations: Historical and Contemporary Population Characteristics.