Fair Use in the United States states says:
...the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use the factors to be considered shall include—
(1) the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
(2) the nature of the copyrighted work;
(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
(4) the effect of the use upon the potential market for or value of the copyrighted work.
Read this handy guide for more info.
To paraphrase my second link: if you're writing something for academic reasons, or your source is in the public domain, then you don't have to worry, you're covered by fair use. For critical commentary or non-profit reasons, as long as you're not illegally reproducing the work, you're fine. The issue is there's no clear cut line for how much of a work it is okay to quote before you cross into copyright violation. Generally though, the shorter the source is, the less you can get away with quoting before you run into trouble.
If you're quoting a book or long scholarly article, then a few paragraphs is probably nothing to worry about as long as you properly cite them and attribute them as quotes in your writing. If you're uncertain though, you can ask permission from the author. The guide I linked has an example of how to do that.
Unless you're reproducing a very large part of a published work though, you probably do not need to ask for permission. Just think of it in these terms:
- Let readers know when you're using somebody else's ideas. Properly cite your quotes and extracts so it's obvious when you're doing this. Highly recommended you use a style guide (MLA, APA, Chicago, etc) so your citations and attributions are consistent and readable.
- Don't use so much of another person's writing that you're basically giving away a copy of it for free with your own writing. If you've quoted / extracted so much of a person's work that anybody who read your book / article would have no real reason to buy the original author's work because they have the bulk of it already in your work, then you've probably crossed a line and need to ask permission to use it.