Tobias summarizes a topic very well. :)|
I suggest you go take a look at Bound by Law?
, which is a comic book explaining many concepts of copyright and fair use for a filmmaker. Many of the concepts can be generalized quite easily. It's produced by the Duke University Center for the Study of the Public Domain and was written by lawyers.
As for "Happy Birthday", I believe (IANAL) that the circumstances under which royalties would be due include any commercial usage, performances in a public place, etc., but does not include singing the song at a birthday party at home.