Under certain conditions, you may copy a copyrighted work without a license from the original author. One of these limitations on the rights granted to the copyright holder is called "fair use." A more restricted version called fair dealing generally applies outside the United States.
Generally, fair use exceptions are ill-defined, and vary widely from country to country. What is fair use in one country may not be in another country.

Under US copyright law, the primary things to consider when asking if something is fair use (set forth in Title 17, Chapter 1, Section 107) are:

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;
4. The effect of the use upon the potential market for or value of the copyrighted work.

Asking yourself these questions might help you determine if something is fair use:

• Is it a for profit competitor or not? Is it for criticism, comment, news reporting, teaching, scholarship, or research? Is the use transformative (of a different nature to the original publication)?
• Is it a highly original creative work with lots of novel ideas or a relatively unoriginal work or listing of facts? Is the work published (to a non-restricted audience)? If not, fair use is much less likely.
• How much of the original work are you copying? Does the portion that you are copying constitute the "heart" of the work and/or its most powerful and significant part? Are you copying more or less than the minimum required for your purpose? The more you exceed this minimum, the less likely the use is to be fair. Are you reducing the quality or originality, perhaps by using a reduced size version?
• Does this use hurt or help the original author's ability to sell it; in particular, does it replace the market for authorized copies? Did they intend to or were they trying to make the work widely republished (as with a press release)? Are you making it easy to find and buy the work if a viewer is interested in doing so?

None of these factors alone is sufficient to make a use fair or not fair - all of them must be considered and weighed. It's routine for courts to express degrees of acceptability or unacceptability for each factor and try to come to a summary and conclusion based on the balance.
Since the 1990s, US rulings on fair use have emphasized the two questions of (1) whether the usage was for a transformative purpose (i.e. a different purpose than the original market purpose) and (2) whether the usage was appropriate with regard to community practice in the community (i.e. higher education) in which the usage took place.

Quotations are a very well known and widely used form of fair use and fair dealing and are explicitly allowed under the Berne convention.

If you produce a derivative work based on fair use, your work is a fair use work. Even if you release your changes into the public domain, the original work and fair use of it remains and the net effect is fair use. To eliminate this you must make the use of the original so insubstantial that the portion used is insufficient to be covered by copyright.

It is possible for a work to be both licensed and fair use. You may have a license which applies in one country or for one use and may make fair use in other cases. The licenses help to reduce the legal risk, by providing some assurance that there won't be legal action for the uses they cover.

Stanford University Library - Summaries of Fair Use Cases
Giglaw - What is "Fair Use" in Copyright Law?
Nolo law center - When Copying Is Okay: The "Fair Use" Rule
Fair use - Center for Media & Social Impact