Fair use is an important part of copyright law that provides some flexibility for users and new creators. At its core, fair use ensures that there are some kinds of uses that do not require permission or payment. But there are no easy rules for fair use - if you want to take advantage of its flexibility, you have to understand its complexities!
Although there are other exceptions to the far-reaching rights of copyright holders, most of those exceptions only apply in very limited circumstances. Fair use is much more flexible, but also much harder to understand and apply. To understand fair use, you need to be familiar with the four statutory factors, and the idea of "transformativeness". To think through whether a particular use is a fair use, you have to look at these details and other associated issues as a whole. Even then, fair use is unpredictable enough that the best anyone can do is make a well-informed, reasonable guess.
Each possible use of an existing work must be looked at in detail and the law spells out several factors that determine whether a use is fair. No one factor is decisive - you always have to consider all of them, and some additional questions. Even after considering all relevant issues, the result is usually in impression that a particular use is "likely to be fair" or "not likely to be fair." There are rarely definitive answers outside of courts.
This is the only factor that deals with the proposed use - all the others deal with the work being used, the source work. Purposes that favor fair use include education, scholarship, research, and news reporting, as well as criticism and commentary more generally. Non-profit purposes also favor fair use (especially when coupled with one of the other favored purposes.) Commercial or for-profit purposes weigh against fair use - which leaves for-profit educational users in a confusing spot!
One element of this factor is whether the work is published or not. It is less likely to be fair to use elements of an unpublished work - which makes sense, basically: making someone else's work public when they chose not to is not very fair, even in the schoolyard sense. Nevertheless, it is possible for use of unpublished materials to be legally fair.
Another element of this factor is whether the work is more "factual" or more "creative": borrowing from a factual work is more likely to be fair than borrowing from a creative work. This is related to the fact that copyright does not protect facts and data. With some types of works, this factor is relatively easy to assess: a textbook is usually more factual than a novel. For other works, it can be quite confusing: is a documentary film "factual", or "creative" - or both? What about the annual "Dance Your Ph.D." contest?
Amount: this is an element that many guidelines give bad advice about. A use is usually more in favor of fair use if it uses a smaller amount of the source work, and usually more likely to weigh against fair use if it uses a larger amount. But the amount is proportional! So a quote of 250 words from a 300-word poem might be less fair than a quote of 250 words from a many-thousand-word article. Because the other factors also all come into play, sometimes you can legitimately use almost all (or even all) of a source work, and still be making a fair use. But less is always more likely to be fair.
Substantiality: this element asks, fundamentally, whether you are using something from the "heart" of the work (less fair), or whether what you are borrowing is more peripheral (and more fair). It's fairly easily understood in some contexts: borrowing the melodic "hook" of a song is borrowing the "heart" - even if it's a small part of the song. In many contexts, however, it can be much less clear.
This factor is truly challenging - it asks users to become amateur economists, analyzing existing and potential future markets for a work, and predicting the effect a proposed use will have on those markets. But it can be thought of more simply: is the use in question substituting for a sale the source's owner would otherwise make - either to the person making the proposed use, or to others? Generally speaking, where markets exist or are actually developing, courts tend to favor them quite a bit. Nevertheless, it is possible for a use to be fair even when it causes market harm.
Transformative use is a relatively new addition to fair use law, having been first raised in a Supreme Court decision in 1994. (Campbell v. Acuff-Rose Music, 510 U.S. 569 (1994.) A derivative work is transformative if it uses a source work in completely new or unexpected ways. Importantly, a work may be transformative, and thus a fair use, even when all four of the statutory factors would traditionally weigh against fair use!
Parody: Parody is one of the most clearly identified transformative uses, but any use of a source work that criticizes or comments on the source may be transformative in similar ways. Legal analysis about this kind of transformative use often engages with free speech issues.
New Technologies: Courts have sometimes found copies made as part of the production of new technologies to be transformative uses. One very concrete example has to do with image search engines: search companies make copies of images to make them searchable, and show those copies to people as part of the search results. Courts found that those thumbnail images were a transformative use because the copies were being made for the transformative purpose of search indexing, rather than simple viewing.
Other Transformative Uses: Because transformative use is a relatively new part of copyright law, it is still developing. Many commentators suggest that audio and video mixes and remixes are examples of transformative works, as well as other kinds of works that use existing content to do unexpected and new things. There is a lot of room for argument and interpretation in transformative use!
It can be very difficult to apply basic ideas about fair use to specific situations. One good way to develop your general understanding of fair use is to discuss examples with others - you'll be surprised how easily reasonable people can disagree!
Since even experts usually only make a "best guess" as to whether a use is fair or not, it's not easy task to think through whether a use you want to make is a fair use.
Links to resources that provide an overview on copyright laws and policies.