I found a free, searchable index of the 1880 U.S. Census at FamilySearch.org. Other on-line Census records exist, but seem to require payment (e.g., a subscription to Ancestry.com).
Is there a way to get free access to years other than 1880?
What’s special about 1880 that it’s free?
The 1880 Census is “special” in that the Mormon Church (which runs FamilySearch.org) actually paid people to transcribe the census returns into a searchable database for their site. Portions of other census records are available on various genealogy web sites, sometimes for a fee, sometimes not. You should also be aware that the census returns for 1890 were destroyed in a fire and are not available.
The LDS (Church of Jesus Christ of Latter-Day Saints) developed the index to which you are referring and chose to make it available for free.
As far as I know, The US 1880 census is “special” because it was the first to be indexed/soundexed by the WPA to assist people in proving their birthdate in order to enroll in Social Security. However, the WPA project only listed families with children under 10 years of age, due to the intended audience.
The USGenWeb Project http://www.usgenweb.org is an volunteer organization dedicated to providing genealogy resources (including census indexes/transcriptions) online at no cost to visitors. As it is a volunteer organization, what is available varies widely.
I would recommend visiting the sites of the state(s) and/or county/ies of interest to see what census material is online. You might even consider volunteering to help with a transcription/indexing project (hint hint).
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What plnnr said.
All the census data extant 1930 and before is freely available offline. Most of the larger cities have at least one library with the complete set of microfilm. The 1930 was only released in the last year, so some libraries may not have that one yet.
Much information from the 2001 Census of Canada is available on the Web. They’ve got it hooked up to clickable maps and a search engine so that you can find a place and retrieve data about it. Neat.