There is no single comprehensive source for American and Canadian newspapers. In some cases you will need to look at one website for current and recent issues of a newspaper eg Nexis and at a different website to look at older issues of the same newspaper. Some websites are free, others can be searched for free but require you to pay to read the articles you find, and others require you to take out a subscription to search them.
Websites to help you find American and Canadian newspapers
Directory of Historical Newspapers on the Internet
Links to various newspaper archives, arranged via state. Includes both fee paying and free resources.
Chronicling America: Historic American Newspapers
contains the U.S. Newspaper Directory which gives publication details of American newspapers published between 1690 and the present, as well as the full text of selected historical newspapers from the period 1836-1922
Pay to use service containing wide range of digitized papers
Archives of individual newspapers:
Brooklyn Daily Eagle Online, 1841-1902
Boston Globe Archive (1/1/1980- ) offers free indexing but you must pay for full text.
ProQuest Historical Newspapers: Chicago Tribune (1849-1994)
Full-text searchable database part of ProQuest Historical Newspapers through FindIt.
New York Times
Keyword searchable archive of articles from the New York Times since 1851.
Note: Articles from years 1923-1986 are not free of charge.