What is copyrightable?
Copyright law is complex and varies by country. The article on public domain at Wikipedia does a reasonable job of explaining what material is copyrightable. See also this table from Cornell. In short, in the United States something is copyrightable unless it was created by the federal government or lacks creative content. Raw facts are not copyrightable. For example, the date that your grandmother was born is not copyrightable. Works involving creativity, such as a two-paragraph biography about your grandmother's life, are copyrightable. A collection of facts is copyrightable insofar as creativity was used to determine which facts to put into the collection. For example, an area phone book is not copyrightable, but a directory of businesses that might appeal to the Chinese-american community is copyrightable.
Brief quotations of copyrighted text (less than 200 words) may be used to illustrate a point, establish context, or attribute a point of view or idea. Quotations of the cited portion of a source are also highly encouraged in the "Text/transcription" field for the relevant source. Copyrighted text that is used verbatim must be attributed with quotation marks or other standard notation, such as block quotes. Any alterations must be clearly marked, i.e. [brackets] for added text, an ellipsis (...) for removed text, and emphasis noted after the quotation as "(emphasis added)" or "(emphasis in the original)". Extensive quotation of copyrighted text, regardless of proper attribution, is prohibited.
Texts published before 1923
Texts published in the US before 1923 are in the public domain and are not under copyright.
Texts published between 1923-1978
Text published in the US between 1923-1978 may be under copyright. Pages containing such text will be marked with the Copypaste template. Material may either be 1) rewritten or shortened; or 2) demonstrated to be out of copyright or licensed. If neither is done within 2 weeks, it will be subject to removal.
Texts published after 1978
Test published in the US after 1978 are subject to copyright. Extensive quotations (greater than 200 words) will be subject to deletion.
Is a GEDCOM file copyrightable?
Textual notes in a pedigree are copyrightable if there is more than one way to express the ideas contained in the notes. The collection of facts in a pedigree may or may not be considered copyrightable. Different people have different viewpoints on this issue.
For the purpose of this licensing discussion we will try to avoid the issue of whether or not GEDCOM files are copyrightable by focusing on biographies and articles, which are clearly copyrightable.
What is Fair Use?
Fair use of copyrighted material is allowed in certain situations under US copyright law. The article on Fair Use at Wikipedia is a reasonable explanation.
What is licensing?
Example usage licenses of other organizations
Below are excerpts from the licenses of a couple other organizations describing how you can use the content found on their websites.
Ancestry.com is protected by copyright as a collective work and/or compilation, pursuant to U.S. copyright laws, international conventions, and other copyright laws. ... You are licensed to use the Content only for personal or professional family history research, and may download Content only as search results relevant to that research. The download of the whole or significant portions of any work or database is prohibited. Resale of a work or database or portion thereof, except as specific results relevant to specific research for an individual, is prohibited. Online or other republication of Content is prohibited except as unique data elements that are part of a unique family history or genealogy.
This site is owned and operated by the Corporation of the President of The Church of Jesus Christ of Latter-day Saints. All material found at this site is owned or licensed by us. You may view, download, and print material from this site only for your personal, noncommercial use or, if you are a professional genealogist, for use by a current client. You may not post material from this site on another web site or on a computer network without our permission. You may not transmit or distribute material from this site to others. You may not use this site or information found at this site (including the names and addresses of those who submitted information) for selling or promoting products or services, soliciting clients, or any other commercial purpose. If you have questions or desire permission to post information, send e-mail to: firstname.lastname@example.org. 
In the following section we describe the GFDL and two alternative "open-content" licenses.
Gnu Free Documentation License (GFDL)
Basically, the GFDL allows anyone to copy, edit, or redistribute material posted on WeRelate, for both commercial and non-commercial use, as long as they:
WeRelate chose the GFDL initially because it is the license Wikipedia uses. An advantage of the GFDL is that is allows content from Wikipedia to be freely cut-and-pasted to WeRelate. A disadvantage is that people wanting to distribute copies of the material on WeRelate have to follow the last two requirements.
Creative Commons BY-SA (CC-BY-SA)
The CC-BY-SA license is similar to GFDL but removes the last two requirements. Instead, they require only that the URL of the CC-BY-SA license page be included in any distributed material.
Creative Commons BY (CC-BY)
The CC-BY license further removes the requirement that enhancements made to the content must be freely shared under the same license.
Both the CC-BY-SA and CC-BY licenses retain the attribution requirement: contributors of the material be given proper credit for their work.