Yes, this is a rather tricky question, and points towards a nebulous conclusion: we can't specify that newspapers as a medium are a primary or a secondary source, rather we need to look at specific articles and decide whether they are primary or secondary. Certainly articles can be primary sources, however, as a general rule, it appears that most newspaper articles can properly be considered secondary sources.
I agree that it may depend on the article, though generally a newspaper would be considered a secondary source. If, for example, an interview is published in its entirety, it could be considered a primary source. As soon as only pieces of the interview are used, it moves to a secondary source because there is no way to determine context without the entire interview.
While newspapers have traditionally been considered primary sources by historians, today's milieu changes things a little bit. Many newspapers carry articles written from syndicated articles and press releases. While foreign correspondent pieces would be primary sources as the correspondent was there at the event being reported, press release story rewrites would be secondary sources. For example, updates and new science reports from CERN about the Large Hadron Collider (LHC) are secondary while the original press release itself is primary as it is written by science experts at CERN and is authentic due to their presence at the LHC. Press releases from which stories are rewritten are usually easy to access from the source referenced in the secondary source story.
The two previous posts are pretty accurate on their definitions of primary vs. secondary sources. Basically, if a reporter witnesses an event and writes about it, it is a primary source. If the same reporter receives the information from witnesses or the police, for example, it is secondary. Many letters to the editor are primary source material, but some, which merely rehash information found elsewhere, would be considered secondary.
Anything that is a first-hand account of people and the way they lived is considered a primary source. Anything that is interpreted is considered a secondary source.
With this in mind, newspapers can be BOTH primary or secondary sources depending on the purpose of the article and how it was written. If it is a factual account, recording the events as they happened, it is a primary source. If the article is interpreted or has opinion interjected, it is considered a secondary source.
Other primary sources include official documents (birth or marriage certificates), diaries, photos, personal calendars, recordings of television and/or radio broadcasts, items collected at events (brochures, ticket stubs, etc.), autographs, maps, artifacts that are handmade and tell something about the culture (jars, etc. that would be dug up in a archeological dig, eyeglasses or pocketwatches from ancestors, etc.).
Secondary sources include biographies, histories, analyses, reviews, books, etc.
Newspaper stories are generally secondary sources of information because they are not first-hand reports describing the action that is creating the news. Most newspaper stories are written by reporters who have gathered information from witnesses or participants in a newsworthy event, so the story's content is secondhand news. If someone who actually took part in the event wrote the story, that would be a first hand resource. If the person who personally climbed Mt. Everest wrote a news story about the climb and the new equipment used in the process, that would be first hand news concerning the climb and the evaluation of the equipment.
This is really a gray area. As you can see from the links below, different people and institutions think differently about this question. I think that it depends to a great extent on the content of the newspaper article.
Some newspaper articles are clearly primary sources. For example, a reporter's account of something that he or she witnessed first hand (like a Congressional hearing) is a primary source. Similarly, an opinion piece in a newspaper is primary.
However, some newspaper articles could be secondary sources. In the New York Times, for example, Nate Silver writes commentaries on opinion polling. These articles are secondary sources because they take original polls and try to analyze them.
So, I would argue that this depends to a great extent on what the content of the article is.
In most cases, they are considered a secondary source. A primary source would be the person the reporter interviewed for the story or a witness who saw the story unfold.