Decius is right. Copyright is, unfortunately, a very complex area of the law. It varies from country to country, it varies by year of creation for many works, and it can be modified overnight if your country signs a treaty with another country or becomes a signatory to the Berne Conventions.

None of which is very helpful to somebody who just wants to make a video.

To make things as easy as possible for yourself, try using music that is definitely in the public domain. Nobody is going to get upset if you take songs on which the copyright has expired. This works better if you play the music yourself or use the song to create a midi file. Some "versions" of songs are in the public domain--like Mariutch on Edison Records. But I'm not sure, for example, that Spike Jone's version of My Old Flame would be public domain. Each song would have to be researched separately.

You CAN ask for permission from the artist. Some artists love this sort of thing, others won't even bother to respond to you. Unfortunately, not all artists own the copyright to their work. Michael Jackson owns the copyright to most of the Beatles works, for example. (And from all I can tell, he owns the copyright to about 2/3 of the music created in the 20th Century.) I've heard Jackson is not easy to deal with on such matters. (Really it's his people I'm sure.)

But this is the reason for all the cartoons that use classical music. Like Elmer Fudd singing "Kill the Wabbit." Wagner is not going to sue for copyright infringement.