Yes, you can usually quote a newspaper article (or a website, or a book) on your blog. Quoting excerpts of protected works is typically considered a “fair use” of the work, and doesn’t require getting permission. Fair use is a broad concept that covers a number of different uses, and quoting an article to engage with it, e.g. through commentary or criticism, is among the most fundamental of such uses.
A classic example of where quoting an article is a fair use would be when a blog post author breaks down the contents of an article or a post on another website, and offers commentary, whether supportive or critical, on the original work. Another example is quoting an excerpt from an article in a discussion forum, or on Facebook, in order to generate discussion among users.
Although we don’t always think of it that way, copyright law places some limits on the First Amendment, which says that “Congress shall make no law… abridging the freedom of speech.” But a strict application of copyright means that you can’t use certain words or certain images when you speak. Limitations and exceptions to copyright, like fair use, are critical safety valves that help balance the incentives to create new works offered by copyright protection with free speech concerns. Protecting the right to comment on, criticize, or parody the works of others is one way that fair use makes copyright law respect the freedom of speech.
If you’re using more of the work than fair use allows, or if you’re using it in a context that isn’t fair use, you still can use the work by getting permission through a license. Sometimes that can be a daunting task of tracking down the copyright owner—but sometimes, copyright holders make things easy by preemptively granting permissions.
For example, if a blog, book, or paper is published with a Creative Commons license, the license may grant you express permission to do more than just take limited quotes. If you want to illustrate your own work with images, for example, you could tap into Flickr’s large collection of Creative Commons licensed work, or access content that might be provided free of restrictions, like the Getty Museum’s Open Content Program or the National Gallery of Art’s NGA Images website.
One final consideration, which isn’t a copyright issue where the original work is text, is attribution. With a very limited exception, attributing a work or linking back to the original work isn’t required for fair use, either as a requirement of copyright law, or in order to qualify as a fair use. However, it’s generally a good practice for a number of reasons. Claiming someone else’s work as your own raises ethical and potentially legal concerns, and it’s just good Web manners to provide your readers with links to sources whenever possible.