Under what circumstances can I use someones copyrighted material without the copyright holders permission?

Copyright law is intended to give creators certain rights to their work so that they can gain recognition and compensation. They have the exclusive right, for the term of the copyright, to reproduce, distribute, make adaptations, and publicly display or perform the work. However, copyright law does allow people to use copyrighted works without the copyright holders permission, in certain cases where the use is limited and serves an important public purpose. For example, the law allows for the fair use of a copyrighted work for such purposes as teaching, scholarship, research, criticism, comment, and news reporting. Whether or not a particular use is fair depends on the consideration of four factors:

The purpose and character of the use, including whether such use is of a commercial nature or is for non-profit educational purposes
The nature of the copyrighted work
The amount and substantiality of the portion used in relation to the copyrighted work as a whole and
The effect of the use upon the potential market for or value of the copyrighted work.

Students, faculty and staff at colleges and universities sometimes believe that any use they make of a copyrighted work falls under fair-use, and doesn't require the permission of the copyright holder, because they are associated with an educational institution. As you can see from the four factors above, that is not the case. In particular, if your use is not for educational purposes, and/or you have downloaded or shared the whole work (i.e. the whole movie, or whole song), and/or your doing so affects sales of that work or the market for licensing fees to use that work, it is highly unlikely that your use would be considered fair use. Put simply, downloading songs, movies, games, and software for your personal entertainment without explicit permission from the copyright owner is not fair use and is against the law.

Additional Information:

Fair Use Checklist