Most of the free newspaper archives that are available are pretty local in nature, covering a particular town, county or state. But a few have a broader national or regional scope:
SmallTownPapers.comThis collection gets bigger every time I look at it. It includes newspaper archives from all over the country, and like the name says, mostly from small towns and cities. Mostly 20th century coverage, but there is a fair amount of material going back the 1800's.
Google News Archives
This is one of the internet's great, underappreciated resources. The content here is a mixed bag -- some results are free to access, while others lead to fee-based services. Still, it allows for fast and comprehensive searching, using the wonderful Google interface. It's a great place to get a feel for what sort of news items might be available, before visiting some more specialized resources to continue your research.
New York Times Archives
I know, I know, it's a New York newspaper! But The NY Times is also a national paper, of sorts, so I've listed here (it's also listed with East Coast newspapers, in case you're worried). Archives from the time the newspaper began publishing in 1851 to 1922 are free of charge, as are articles published after 1987. Everything in between (1923 to 1986) you have to pay for, for some odd reason.
NewspaperArchives.com
HistoryBuff.com
This small but intriguing collection generally has only one or two archived newspaper pages for any given year. But it manages to cover most of the major events of the 18th, 19th, and 20th centuries. Unlike most archives, the page images are especially crisp, and photographs, in particular, are very clear. There is no text searching, but you can browse by date.
Centennial Newspaper Exhibition, 1876
For newspaper buffs, this is a book worth noting...a complete listing of every paper published in the United States at the time of publication, along with circulation and demographic statistics. The entire book can be downloaded, free of charge. Thanks to U Mich for making this available.
Other pages at this site: