Fair use is the most relevant copyright exception for students since it allows many uses of copyrighted works for the purposes of education and research. However, not all educational uses are fair uses.
Fair use is a doctrine found in US copyright law that recognizes that there are certain uses of copyrighted works that don't require permission from the author or copyright holder. Some of the "classic" examples of situations where a use is likely to be fair include criticism, commentary, parody, news reporting, teaching, scholarship, and research.
Everyone is free to exercise their fair use rights, but in most cases, only a judge can definitively decide if a use is fair or not. That means that those who want to exercise their fair use right will need to do so with careful thought and will need to weigh the risks involved in certain uses.
Fair use is purposefully vague so that it can remain flexible as technologies and types of uses change over time. However, its vagueness can make it more difficult to decide when your use is or isn't fair. In order to decide if a use is fair, you have to analyze the particular details of your situation and proposed use.
Thinking Through Fair Use from the University of Minnesota Libraries, the Fair Use Evaluator from the American Library Association, the Fair Use Checklist from Columbia University Libraries are all helpful tools to use when you are analyzing your proposed use in light of the fair use doctrine.