A census is an official counting of a state or country's citizens, used to gather demographic information about the population. The United States has been taking a census every 10 years since 1790. You probably filled out a census form that came in the mail last spring for the 2010 census, which asked you to provide certain information about the people in your household. Your ancestors did the same thing, every 10 years... and looking at their census records can give you a lot of information about them!
Each census asked different questions, depending on what type of demographic information the government was interested in at the time. The first census, in 1790, came shortly after the Revolutionary War, when the big concern was, "If we have to raise an army again, how many men of fighting age can we count on?" So this first census was very basic: it recorded the name of the head of household, and then tallied the number of males over 16, number of males under 16, total number of females, and number of slaves, if any. The 1800-1840 censuses were similar: only the head of household was recorded by name, and the other household members were recorded as tally marks under age and gender.
In 1850 and onwards, they did away with the tally marks and recorded the name of every household member, along with their age, gender, birthplace, occupation, and other info. In 1880 onwards, they asked not only for each person's birthplace, but also for the birthplace of each person's father and mother. In 1900, they asked even more genealogically-helpful questions: year of immigration and naturalization status; how many children each woman had given birth to, and how many were currently living. In 1930, they asked if the family owned a radio!
Here's a sample of the 1850 census:
http://lh3.ggpht.com/_3DAJUNADSXE/SmwPfz77n0I/AAAAAAAA4co/gUDOWfmmgHs/1850+census+isaac+and+elizabeth+taylor+and+kids+part+2.jpg
And here's a sample of the 1900:
http://www.recordranch.com/genealogy/Images/LammersJohn1900Census.jpg
You can see all the questions for each census right here:
http://www.ancestry.com/charts/census.aspx
To protect the privacy of living people, census information is kept private for 72 years. We currently have access to the 1790-1930 census; next spring, we'll get to see the 1940 census for the first time.
Where to view the census online:
Ancestry.com
(has the complete set of 1790-1930 census images online)
Familysearch.org
(has images for 1850-1900... 1910-1930 will be available soon)
HeritageQuest, available through libraries, also has census records.
Cyndi's List has an extensive listing of other websites with census records:
http://www.cyndislist.com/census.htm