Copyrights are very important because they identify who actually owns the song and song recording and who gets to make money from it. When songwriters write songs, the songs are automatically copyrighted as soon as they are in a tangible form (like a recording, or fixed as printed sheet music). In order to sue for copyright infringement, however, the song should be registered with the copyright office at the Library of Congress. Registration should always be done before the song is set loose in the public domain (available to hear on a Web site, etc.).
There are several separate licenses involved with music because different parties can own different parts of copyrighted material. There is the composer who owns the copyright of the composition. There is the publisher who can grant usage rights of the composition. The Performing Rights Organizations (PRO) usually collect performance royalties on behalf of the composer and publisher (unless a Direct License has been issued to the Licensee) when applicable. The sound recording is also copyrighted, which can be owned by a recording company.
Since a music library typically owns the copyright on the sound recording and the publishing rights, all these licenses are easy to obtain from one source in one transaction. Performing rights for a production music library can either be handled by the PROs, or obtained directly from the music library in what is known as a Direct License. Many small royalty free music libraries offer the direct license as a standard part of their music license. Other music libraries may grant a direct license for certain uses while requiring the PROs to handle others. Some music libraries do not grant direct licenses under any circumstance. This will vary between libraries.
When licensing so called "popular music", the costs are far greater and the various licenses have to be obtained from different sources. You need to contact the music publisher (or composer if no publisher) to obtain mechanical rights, synchronization rights and inquire about performing rights. Then, you need to contact the record label to obtain the master use rights as they own the copyright on the sound recording. Contracts for each right will need to be filled out with each party involved.
Another type of license worth mentioning here, which has nothing to do with an Audio/Visual Work or Audio Only production, is Grand Rights. These are the rights needed to use music in a broadway style performance, and are granted by, and paid directly to, the music publisher.