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.
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.
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 Is the work published or unpublished?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.
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.
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.
