|Some sort of strange census machine|
(photo courtesy of Marcin Wichary, CC-BY)
I'm guessing that the plural of census is either the awkward-sounding censuses or the pretentious-sounding censii, but I'm too lazy to look it up, so I'll just roll with census records.
There are a ton and a half of state census records available online, mostly name searchable and all free of charge. Some records go back right to the days when the state was first settled, others focus on particular sub-populations like Indians or war widows and, of course, many are general census records for the entire population.
Without further ado:
The Kansas Historical Society Name Index includes the 1895 state census, and a bunch of other good stuff, including a directory, of sorts, of pioneer women.
The Maryland State Archives Census Indexes cover 1776, 1778, 1870 and 1880, with straighforward name lookups.
The Idaho State Historical Society houses the oddly-named Idaho non-population census for 1870 and 1880 which recorded workers in specific industries, along with 1890 Idaho state census. These are PDFs that you can download and search. You'll find a lot of Chinese names here, reflecting the immigrant workers in the mining industry.
Minnesota seems very keen on census-taking, as the Minnesota State Census Index includes records from ten censuses/censii between 1849 and 1905. Your online search provides preliminary information, and you can then order a full copy of the census record for a fee.
The Washington State census collection has a zillion online censuses from 1847-1910, but that's because they're listed by county. Happily, you can search them all at once at the Digital Archives Search.
New Jersey's online archives have some very cool records, including the N.J. 1885 Census and a very unusual tabulation of 100 years of legal name changes...how often do you see that!
The North Dakota State University Archives includes the Dakota Territory 1885 Census where you can look up names and order census copies. Take a look at the other databases available on the left side of the page.
Search the Colorado Census of 1870, which is actually the excerpt from the federal census. There are quite a number of other historical records here as well.
There are doubtless online census records in other states as well, but this is as far as I've gotten in cataloging them. Stay tuned for future updates...
Don't forget to also check for your family history at Ancestry.com and NewspaperArchive. These are subscription databases, but they are among the most powerful research tools available for looking into family roots.