This is the complete "head-count" of the U.S. population conducted every ten years, primarily to determine representation in the U.S. House of Representatives.
Original printed volumes of the decennial census are available in Leyburn Library's U.S. Government Depository collection, but many of the older volumes are not in good physical shape.
Reprinted volumes of the first ten decennial censuses (1790-1880) are shelved in the Reference collection.
Below are links to digitized copies of decennial census volumes and related materials from various government and university projects.
This is the Census Bureau's own project to digitize (PDF and ZIP files) past volumes of the decennial census, 1790-2000.
Scans of "all available published [decennial] Census volumes in PDF format," 1790-2000. From the University of Minnesota's IPUMS-USA site, which also contains facsimile copies of the enumeration forms used in the decennial census of the United States from 1860 through 2000.
Data from the 1790-1960 decennial census counts, compiled by UVa's GeoStat Center.