In some cases you can use a work that is not in the public domain without seeking permission. This is known as "fair use." The problem with fair use is that there is no way to be certain that yours is a fair use until a court decides. However, like all rights, if you don't exercise fair use, then eventually you'll lose it.
Below are a few questions that you should ask yourself when trying to determine if your use of an image would be considered fair or not. These "four factors" should be weighed against each other; no single one counts more heavily than another. In general, using a small portion of another's work for educational purposes is considered fair.
1. What is the character of the use? Is it for educational use or to criticize/parody, or do you hope to make some money? Courts usually favor educational uses over commercial ones.
2. What is the nature of the work to be used? Is it primarily factual, or imaginative? Is the work published or unpublished? Courts tend to rule against uses of creative works and those that have not yet been published.
3. How much of the work will you use? Only a small amount, or a significant amount? Amount is measured both quantitatively and qualitatively. If you're only using a small portion of a work, the court is more likely to rule in your favor.
4. If this kind of use were widespread, what effect would it have on the market for the original? Would the copyright owner be losing money? Courts usually favor those uses with little or no adverse market effect.
For more on fair use, see the University of Texas' Four-Factor Guide.
Digital Images and Copyright (from Colgate University)
A brief guide to using digital images from the internet (and in general)
Best practices in fair use of dance related materials (Annotation from the Visual Resources Center, University of Texas, Austin)
The Dance Heritage Coalition has released a “Statement of Best Practices in Fair Use of Dance-Related Materials: Recommendations for Librarians, Archivists, Curators, and Other Collections Staff.” Although specific to dance, the report is an excellent resource for information about copyright in general and fair use in particular of performing arts materials, both by patrons and especially by librarians in creating exhibitions and websites, format migration and preservation, and public performance.
An interactive program from the Visual Resources Association designed to assist users in determining the copyright status of a specific digital image
In its own words: "This Statement of Best Practices in Fair Use makes clear what documentary filmmakers currently regard as reasonable application of the copyright “fair use” doctrine. Fair use expresses the core value of free expression within copyright law. The statement clarifies this crucial legal doctrine, to help filmmakers use it with confidence. Fair use is shaped, in part, by the practice of the professional communities that employ it. The statement is informed both by experience and ethical principles. It also draws on analogy: documentary filmmakers should have the same kind of access to copyrighted materials that is enjoyed by cultural and historical critics who work in print media and by news broadcasters."
Extensive information and examples of fair use cases from Stanford University.
Basic information about copyright from the government
The Visual Resources Association is comprised of members that professionally digitize and create access to images for educational use. The Statement describes six uses of copyrighted still images that the VRA believes fall within the U.S. doctrine of fair use. The six uses are: 1) preservation (storing images for repeated use in a teaching context and transferring images to new formats); 2) use of images for teaching purposes; 3) use of images (both large, high-resolution images and thumbnails) on course websites and in other online study materials; 4) adaptations of images for teaching and classroom work by students; 5) sharing images among educational and cultural institutions to facilitate teaching and study; and 6) reproduction of images in theses and dissertations. This statment was written by VRA's Intellectual Property Rights Committee with the guidance of a Legal Advisory Committee of preeminent copyright scholars and legal experts.