Sites that allow you to use their content say so explicitly and will detail the terms of use someplace on the site (or point to a license like Creative Commons, GNU free documentation, etc). Some examples are Wikipedia (as Nick pointed out), "e-zine" article banks and most US government sites.
There is also something known as "fair use" - this pertains to using a
short quote of someone else's work for things like reviews, commentaries, news reports, etc. There is no strict legal definition of "fair use". You can read what the US copyright office has to say about it
here
