If you have determined that your use is not covered by any exceptions (Classroom Use, Fair Use, TEACH) and that the work is not in the public domain, you will need to ask the copyright holder for permission. Columbia University has some very helpful model letters you can use as a template.
When asking for permission:
Seeking permission can be a long and frustrating process (though it isn't always) but it is necessary in instances where your use is not permitted by any other part of the copyright law. Remember that not all educational uses are fair uses and that attribution is not a substitute for permission, when permission is necessary.
A frustrating reality is that sometimes even when we want to ask for permission - even to pay for permission! - it can be challenging to figure out who owns a copyright. Because copyright is automatic, requires no notice, and can be sold, given away, or otherwise licensed, it is often the case that the material that we are looking at has no notice of copyright or that the notice of copyright that it has is no longer accurate (as when a publisher is acquired by another publisher or when rights revert back to the original author per a contract).
When the owner is a person
Try to find contact information by searching for them online. Though your initial conversations might be over the phone or on Facebook or Twitter or email, remember that you'll need a formal signed letter of permission in order to really feel confident legally speaking. Remember that people very often do not own the copyright on their work but it is instead own by a publisher or other content producer.
When the owner is a company
Look on their website for a permissions or copyright department. Some companies will have big copyright permissions operations and make it easy for you, some will have little info and you'll simply need to send a letter off to their main address.
When you don't hear back
In our experience, it is pretty common to send a request off and never hear back or to hear back months later. If you do not hear back and your use is really not fair or covered by another exception, the sad truth is that you probably simply cannot use that material.
One of the most vexing problems in copyright law is the problem of "orphan works". These are those great many works where it is difficult or impossible to determine who owns the copyright. The copyright term in the US is very long and copyrights outlive their creators by 70 years. Works by corporate or anonymous authors are covered by copyright for up to 120 years. That is a lot of time for the details of ownership to get lost in the shuffle.
When dealing with an "orphan work", you must basically decide to either take the risk that a copyright holder might later identify themselves or forgo using the work. It is hard to say how risky a given use might be. If a copyright holder came forward they might simply insist that you stop using the work or they may attempt to recover damages.