Note:  Obtaining permission should not be the first step.  In fact, when solving a copyright problem, it is best to follow these steps:

1. Determine whether the work is indeed protected by copyright (is it in the public domain?);
2. Determine whether an exception to copyright applies;
3. Determine whether there is a valid argument for fair use;
4. If the work is protected by copyright, there are no exceptions and fair use does not apply, seek permission to use the work.

In order to obtain permission to use the work, you must determine who holds the copyright for the particular work. You can begin the process by (if the work is a book or a journal) determining who the publisher of the book/journal is. Contacting the publisher to ask permission is a reasonable next step (if the publisher is still in business, they likely know who owns the copyright and they may even own the copyright themselves). Another reasonable step to take to determine who owns the copyright of the work is to search the US Copyright Office page for that information. You may search the US Copyright Office page for information about a particular work by title of the work, name (of the author or publisher), keyword or registration number (if you know it). If you cannot locate the publisher, you may have found an "orphan work." 

If the work is a musical composition, begin with the author of the composition. You could try looking up the record of copyright on the US Copyright Office website as well. You could also contact the American Society of Composers, Authors, and Publishers (ASCAP) to seek permission.

If the work is a sound recording, the job of locating the current owner of the sound recording copyright may be difficult, as these rights often get assigned quite frequently. But, contacting The Harry Fox Agency or Broadcast Music, Inc. (BMI) is a good place to start, as they hold many of the sound recording copyrights in the music industry.

If the work is an audiovisual work and you want to show it in a classroom, as part of your curriculum, see the teaching exception to the Copyright Act. If you want to show a movie for a public showing at a student club or an extra-curricular event, you need to obtain performance rights to show the movie. Try contacting one of the following agencies to obtain performance rights: 

Kenneth Crews has put together a guide for how exactly to ask for permission to use a copyright-protected work and what to include when asking for permission.

IMPORTANT:  If your use is non-profit and/or educational, make sure you note that in your request for permission.