Works Made for Hire
If you've written a song as a part of your job (maybe you work for an ad agency and have written a song for a commercial), you don't own that song -- the company does, because you wrote it as part of your job. This is also true if you've been commissioned to write something as part of a collective work. In this case, however, the agreement you sign will specifically state that it is a work "made for hire."
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.).
As copyright owner, you have the right to reproduce the copyrighted song, to create derivatives or variations of the song, to distribute it to the public, to perform it publicly, and to display it publicly. (Although we're not sure how you "display" a song.) If you have recorded the song with yourself as the artist, then you also hold the sound recording copyright (a different animal entirely) and have the right to publicly play or "perform" that recording by means of a digital audio transmission.
By giving someone a license, you are giving him permission to use your song. Once the song has been recorded and publicly distributed, however, compulsory licensing kicks in and everyone who wants to cover (record) the song can do so without your specific permission. They are required by law to pay you a statutory royalty rate, however, as well as notify you that they're going to release it, and send you monthly royalty statements. They are NOT allowed to make any changes to the words or melody or change the "fundamental character of the song" without the copyright owner's approval. If the song is changed, it is considered a "derivative work." Record companies rarely use compulsory licensing because they don't want to have to provide monthly royalty statements. Instead, they go to the copyright owner and get a direct license so they can negotiate the terms more freely.
If you write the lyrics to a song and your buddy writes the music, then you each own 50% of the song. You don't own all of the lyrics and your buddy doesn't own all of the music -- you each own 50% of the total song, music, lyrics and all. This means you can't give someone exclusive rights to the song on your own if you have a fight with your buddy. And, if you make any money on the song, half of that money must go to your partner.
Other forms of shared copyrights come into play when you or your publisher (typically you give control of the song's copyright to the publisher) sign over a portion of the copyright to another publisher for a sampled composition -- a song that uses a portion of another song.
Transfer of copyrights
In most music publishing agreements, there is a requirement that the songwriter assign the copyright of the written song to the publisher. This is known as a "transfer of copyright," or simply "assignment." This, in effect, transfers ownership of the song to the publisher in exchange for the payment to the songwriter of royalties in amounts and time intervals agreed upon in the publishing contract. Typically, song copyrights are held by the music publishers, while sound recordings are controlled by the record companies.