(We've moved to: http://christinesleeter.org/digitized-historical-newspapers/)
Until the last few decades, newspapers and magazines were the primary forms of media. Not only did cities publish their own newspapers, but small towns did so as well, with probably more local news than is the case today. For research on family and local context history, then, old newspapers, which increasingly can be found in digitized form are a goldmine.
For example, in the process of researching some of my family in central Illinois during the 1800s, I discovered that the town of Decatur -- the largest town in the vicinity -- published several newspapers, some daily and others on weekends. The newspapers carried national news as well as local news, including news from outlying villages. As a result, I've been able to find numerous stories of everyday life in the village where my ancestors were, and several stories include them by name.
Availability of newspapers, print or digitized, depends on what has survived. Due to the Civil War, for instance, more has survived in the North than in the South. Nonetheless, it's worth starting your search by assuming you will be able to locate newspapers from the past, because you may be surprised what you will find.
No one source has everything; I recommend checking multiple sources. If you don't mind paying a subscription fee, Newspaperarchive and Genealogy Bank are useful. You may be able to find much, however, by beginning with free sources.
Family Search, while it does not contain its own set of digitized newspapers, offers a link to free digital historical newspapers in several countries. For free access newspapers, this is a good place to start.
Chronicling America is an open-access joint project of the U.S. Library of Congress and the National Endowment for the Humanities that has digitized newspapers from around the U.S. between 1836 and 1922. The search process is well-labeled and intuitive. You can enter a name or descriptor, and narrow down (if you wish) by state, year range, ethnicity (all, African American, Indians of North America, Irish, Jewish, Pacific Islander, and Jewish are the choices), and language (all, Choctaw, English, French, Hawaiian, Japanese, and Spanish are the choices).
The Directory of Historical Newspapers on the Internet is a useful tool for finding some available sources of digitized newspapers state by state. You can also enter into Google the name of a state plus the phrase "historic newspapers." States vary widely in what is available, but the trend seems to be placing more and more surviving historic newspapers on the Internet. A few states now use Olives' ActivePaper Archive, which I particularly like, perhaps because I've found detailed information about people in small towns in Colorado (the state of Colorado uses this service) and Quincy, Illinois (the Quincy Public Library uses this service.)
If you have links that readers might be interested in, please send them along and I'll incorporate them into another posting.