A classroom presentation or paper illustration?
In general, images used in a classroom presentation, for a scholarly lecture, or in an unpublished assigned paper, fall under the concept of Fair Use or the TEACH ACT. Fair use is an exception to the exclusive rights granted by copyright. For further information, consult Circular 21 of the United States Copyright Office.
A lecture to a paying audience?
If your audience is paying to see you, in general you should obtain permission before using an image, unless the image is in the Public Domain.
A published scholarly article, book, dissertation or website?
You are responsible in these cases for obtaining permission, unless the work is in the Public Domain. In most cases, your publisher will require that you do so.
A new work of art based on another person's art?
Not an easy question to answer. The creator of a copyrighted work of art is given the exclusive right to prepare derivative works based on that work. On the other hand, there is a long tradition of artists responding to others' works. Whether you are basing your work on the original work of art, or a digital or print copy of it, it is always safest to get permission first.
Planning to perform a play, or add some commercially recorded music to your own theatrical work? Check out the links below for some guidelines on how to obtain the necessary permissions.