Here are a couple places that you may want to check out that have royalty-free music that you can use:
ccmixter.org/ - ccMixter is a community music site featuring remixes licensed under Creative Commons where you can listen to, sample, mash-up, or interact with music in whatever way you want.
freesound.org/ - The Freesound Project is a collaborative database of Creative Commons licensed sounds. Freesound focusses only on sound, not songs.
audiofarm.org/ - Audiofarm is an audio website for helping people explore, share, and enjoy Creative Commons Audio from around the world.
At this website Kevin MacLeod offers his royalty free music creations under the Creative Commons License. There are many different types of instrumental music at this site.
pacdv offers a few instrumentals for you to use royalty free. They just ask that you mention "Music bypacdv.com/sounds/" in your credits.
Offers a few songs that you can use in your productions as long as you credit the songwriter - Peter John Ross.
Offers a few royalty free instrumental songs under the Creative Commons License. Just credit the musicians.
Taylor Hayward offers a few piano songs that are copyright free so you are free to use them in any way, commercial or non-commercial.
I recommend that you sign up with a free account atsoundclick.com to take full advantage of its many differnent music artsists and genres. To find royalty free music at no charge under the Creative Commons License click on the soundclick link above and then change License Type to Creative Commons (Free), select the Genre and click go.
This site includes many different genre's of music by many different artists under the Creative Commons License. The difficult part at this site is just downloading one song. When you go to an artist's page and click on an album it wants you to download the whole album. But if you are listening to the album with the jamplayerweb there is an option at the bottom to Export this playlist: M3U. If you pause the music and click on M3U then the songs should open up in your Media Player and you can click the next and previous buttons in your Media Player and also click on File -> Save As... to save the song you are listening to.
Derek Audette offers his royalty free music under the Creative Commons License. Most of these are instrumental pieces that are either dark, moody, or hard driving.
This site contains recordings of songs that were published prior to 1922 and are now in the public domain which means the public is free to copy and use the works in any way.
This website offers a collection of music from a variety of artists and genres under the Creative Commons License. In the left hand menu click on artists or genres.
Open Music Archive offers recordings of songs that are over 50 years old so their copyright has expired in the UK so the recordings have become Public Domain. However, if you are from another country other than the UK then the recordings may not be Public Domain yet. See the FAQ for more information.
This Audio Archive contains free recordings of many popular bands. These are usually recordings of live performances. The bands that allow their live concerts to be recorded and traded are known as Trade Friendly Bands. For a list of bands that are Trade Friendly and to view their policies see Trade-Friendly Band Information and Furthurnet.org Band List. The FAQ page and the details page make it clear that these recordings can only be used for non-commercial purposes.
Free Production Tracks has a few styles of music that can be downloaded for free and can be used for anything, commercial or personal, without any royalty fees.
Royalty Free Music royaltyfreemusic.com/free-music-resources.html
royaltyfreemusic.com mainly offers royalty free tracks for a price, but they do offer some 30 second music clips, beats and loops that can be used for personal or non-commercial productions. You must credit RoyaltyFreeMusic.com as the source of the music.
InernetAudioGuy.com usually sells royalty free audio tracks for a price but they also offer a few instrumental 30 second samples that you can use for free for anything you want.
Using Non-Royalty Free Music
If you already found a piece of music from a professional artist that you want to use in your film or other production then go to ascap.com/filmtv/faq.html to find out how to contact the copyright owner (usually the publisher) and acquire the appropriate license.