Before forging ahead with a permission request to a copyright holder, it's worth examining whether permission is needed.
Is the work in the public domain?
"Public domain" works are either no longer protected by copyright or in a category of work that isn't protected by copyright. Public domain works can be used without obtaining permission.
A work is in the public domain if its copyright has expired. All works published in the United States before 1923 are in the public domain. Copyrighted works created in the United States after 1977 are protected for the life of the author plus 70 years so there are no works from this period that will be in the public domain unless a copyright holder has explicitly stated that he/she has rescinded copyright. (Creative Commons licenses have become an easy way for authors to clearly register that they wish their works to be in the public domain.)
Works published between 1923 and 1978 may be protected by copyright if they were published with a copyright notice and the copyright was renewed. Copyright notice consists of the year, the word "copyright" or the copyright symbol, and the name of the claiming copyright owner. Investigating whether a work was copyrighted in the first place or has been renewed can be a time-consuming process. "How to Investigate the Copyright Status of a work", published by the Copyright Office is a good guide to the process.
Note: As mentioned elsewhere, an author acquires copyright ownership once he/she creates a tangible work in a fixed format. However until that work is registered the author cannot sue for infringement. Again, there is no financial risk of using the work of another author, fair use or not, if that work hasn't yet been registered.
Works created by U.S. government agencies and departments are not copyrighted and are in the public domain.
Copyright durations in other countries differ from those in the U.S. and public domain is defined differently. This public domain calculator is designed for investigating copyright durations in European countries. A web search should be sufficient to learn about copyright durations in specific countries.
Depending on the nature of your intended use and the amount of a work you plan to use, you may be able to comfortably use a portion of the work without obtaining permission by using the Fair Use guidelines. Look at the tab labeled "Fair Use" for information about the four factors that are used to measure what is a fair use. Seeking permission when it isn't necessary is time-consuming and it may lead to a request for licensing fees or a refusal of your request. It can be safely assumed that most copyright holders won't take the time to inform you of the Fair Use guidelines. That is not to say there isn't a risk that your assessment of fair use is wrong. Each case must be considered individually as there are no simple quantified rules about what is and isn't a fair use.
Locate the copyright holder
For print publications, it is most likely that the publisher holds the copyright. The Copyright Clearance Center (CCC) is a gateway to many publishers and can be used to purchase rights by credit card in many instances. They will also assist in seeking permissions if they don't have a given publisher listed. In the United Kingdom, the Copyright Licensing Agency provides similar services.
If you are unsuccessful using the CCC, do a web search for the publisher. Publisher sites usually have information about rights and permissions with contact information.
If the publisher does not own the rights, you may be referred to the author or his/her estate. Be aware that if a work has multiple authors or contributors associated with the work it may be neceesary to secure permission from several parties including co-authors, photographers, artists and others.
Request permission in writing
Plan ahead and allow adequate time for a reply. Receiving permission may require payment of fees. Obviously no reply doesn't imply that getting permission isn't necessary. Be persistent.
Columbia University's Copyright Advisory Office provides several sample letters that can be used as templates for requesting permissions from publishers.
Acknowledge the author(s) when permission has been granted
The copyright holder may have specific instructions on how to credit their work but generally, one should include a footnote in the text; a source note to a table; and/or a credit line under an illustration or photo.
Copyright Crash Course: Getting permission (University of Texas)
Permissions (Columbia U Copyright Advisory Office)
The Basics of Getting Permission (Stanford University)
U.S. Copyright Office - Works registered or renewed after 1976, or reclaimed under GATT, can be researched on the U.S. Copyright Office site
Copyright Term and the Public Domain in the United States (an indispensable table of public domain information created Peter B. Hirtle at Cornell University)
Copyright law and Public Domain (nolo.com)
Public domain collections: Free to share and reuse - from the New York Public Library