According to the U.S. Copyright Office,
Fair use is a legal doctrine that promotes freedom of expression by permitting the unlicensed use of copyright-protected works in certain circumstances.
What this means is that there are numerous exceptions that allow a person to use and manipulate copyrighted works without the owner's permission as long as the user does not violate "fair use" doctrine. Using copyrighted material for the purpose of research, education, criticism, and parody is often considered "fair use."
It is important to note that "fair use" is not an explicitly defined right and only exists as a legal defense. If a copyright holder sues you for copyright infringement, your interpretation of fair use may or may not differ from a court of law's.
17 U.S.C. § 107 gives us some insight as to what considerations the law takes when determining whether the use of copyrighted material can be deemed "fair use".
What are you trying to accomplish by using the work? If you are using a work for non-profit, non-commercial educational purposes, such as instruction or research, use of the work is probably deemed fair. For-profit educational institutions are much less likely to receive sympathy from courts.
Creative factors weigh heavily in determining fair use. Using works of non-fiction or informational writing is much more likely to be fair use than a more imaginative work, such as creative writing or an original song. Using an unpublished work is rarely considered fair use.
Are you using a chapter or excerpt from a book, or close to the whole book? Generally speaking, the more of a work you use without permission, the less likely a court is to rule fair use. This is not a rule set in stone; there have been rulings where use of an entire work was deemed "fair" and use of small excerpts as not fair.
Is your use of the work negatively impacting the author or publisher's bottom line? Will your work hurt the sales of the work that you are borrowing? If the answer to either of these questions is yes, a fair use ruling is unlikely.