Several online sources of information on copyright, permissions, and fair use can help you determine when you need permission to use other people's material in your work and how to go about getting it.
The Association of American University Presses (AAUP) has a Copyright and Permissions Resource page, with a useful Permissions FAQ.
The Stanford Copyright and Fair Use site provides an overview of the permissions process and good discussions of public domain and fair use.
The College Art Association (CAA) has developed a Code of Best Practices for Fair Use in the Visual Arts that provides a set of principles to consider when determining whether you can make use of copyrighted work under the fair use doctrine. The report's introduction and the section on analytic writing are especially helpful for art historians.