Michigan State University has compiled a long list of historic and current newspapers, complete with links to each database. Some of the databases are available free of charge while others are hosted at commercial sites that charge hefty fees.
Hint: Many local libraries subscribe to the commercial databases, such as ProQuest, Gale, and others. You may be able to obtain free access at your library. Some libraries even offer in-home access to these databases by first logging into the local library's web portal.
Some of the databases listed include 19th Century U.S. Newspapers via Gale, America's Historical Newspapers - 1741-1922, Civil War: A Newspaper Perspective via Accessible Archives (more than 2,500 issues of The New York Herald, The Charleston Mercury and the Richmond Enquirer, published between November 1, 1860 and April 15, 1865), and many, many more. The collection includes newspapers from large cities (New York Times - 1851-2009) and small towns (Indiana Farmer - 1851-1917) alike.
The Michigan State University Library Guide also includes links to many foreign newspaper collections, campus newspapers, and ethnic newspaper collections. The ethnic collections include such things as the African American Newspapers, 1827 - 1998 from Readex, Archives Unbound: Japanese American Relocation Camp Newspapers 1942-1945 from Gale, Hispanic American Newspapers - 1808-1980 from Readex, and the Pittsburgh Jewish Newspaper Project.
You can learn more about the Michigan State University Guide to Historical Newspaper Databases at http://libguides.lib.msu.edu/newspaperguide.