Generally, under copyright law, one has to get the permission of the rights holder (a license) to use their original work. However, there are factors under the Fair Use statute in U.S. copyright law that allow for the unlicensed use of copyrighted material.
While more factors may come be considered, the four factors made explicit in the Fair Use statute are:
It is important to know that Fair Use is a defense against claims of copyright infringement. It is up to the court to decide if the unlicensed use is "Fair Use" or if the use violates copyright. No one factor is necessarily more important than another - it depends on the courts interpretations of each case.
Transforming a work to use it in new and unexpected ways can also be considered Fair Use.
Parody is the most widely recognized type of transformative use but the rise of the internet continues to push the boundaries of what is considered transformative. For example, the digitizing of books for the purpose of data mining (and not reading) has been considered transformative. However, what may be considered transformative depends on the medium and the culture surrounding that medium. Music sampling and remixes may be transformative, but currently those uses have not been deemed protected by Fair Use in court.
Public domain generally refers to works with expired copyright, though some creators choose to publish original work free of any copyright restrictions. Works in the public domain may be freely used.
When a work's copyright term expires, there are no longer legal restrictions placed on how it can be used. In the United States, any works published 95 years ago (or earlier) are in the public domain. Additionally some types of government documents and other sources are automatically in the public domain.
These days general rule of thumb for copyright terms in the U.S. is the life of the author plus 70 years, though there are numerous exceptions. When in doubt, check with an expert like a librarian.
A tool to designate the creator relinquishes any inherent rights to original work; the CC0 license is a way to clarify the creators intent no matter the copyright laws of different locations and jurisdictions.