Just be aware that some sites will say "copyright free" and this is an immediate warning signal. "Royalty free" is indeed what you want, as you pay an initial fee and are allowed to reuse said music for certain projects (technically you're buying a license to use the music, so check that the license suits you and your purpose or you could end up owing more money)
There are some companies that sell music for purposes like this, and more advanced stuff like TV and dvd projects. Crocodile, Getty and other companies who advertise under the banner of "production" or "library" music are worth looking into. *
There are two other options:
1) if you know any bands or musicians you could ask for written permission to use their music (as long as its original and not a cover). Try not to spam various bands on Myspace or whereever as they tend to tell you to bleep off though
*
2) This is my method - get some cheap music software (I use ableton Live, but there are cheap versions of Live, Cubase etc) and make your own. You could even import a midi file of some classical, etc (again make sure its royalty-free and not from a source that is likely to get you in trouble) and make a tune like that. Obviously the latter isnt for everyone.
* if you use these options you must write "music (c) copyright whoever year" on the video itself and all accompanying text
