Fair Use is probably the most confusing part of the copyright law due to it's lack of clear quantified guidelines. It's a concept in U.S. copyright law that recognizes that some uses of copyright-protected works do not require permission from the copyright holder.
As quoted from the law (see Title 7, Section 107):
Notwithstanding the provisions of sections 106 and 106 A (on the exclusive rights of copyright owners), 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 —
The fact that a work is unpublished shall not itself bar a finding of fair use if such finding is made upon consideration of all the above factors.
There has been much written on Fair Use and it's recommended that you follow the links provided on the left column to develop a better understanding of the factors.
Copyright and Fair Use: Common Scenarios NYU Libraries
Fair Use Evaluator a tool created by Michael Brewer for the American Library Association. This site guides the user through each of the four factors considered in Fair Use evaluations and creates a time-stamped PDF document for your records.
Summaries of Fair Use cases Stanford University
Thinking Through Fair Use an analysis tool from University of Minnesota
Fair Use and YouTube mashups and remixes - Appropriation of copyrighted footage and images for reuse on YouTube and social media is widespread. Here are some educational tools to inform creators about responsible fair use of copyrighted content in online videos and memes. Copy This podcast; YouTube statement on Fair Use; New Media Rights webpage; Everything is a Remix: Fair Use (video)