I have a concept in mind for a website using HTML and Flash and a little PHP.
The problem is that the flash header up top keeps refreshing every time someone
clicks on a navigation link in the navigation bar. So the intro to the header
keeps replaying every time a new page loads. How do I make it so that when
people browse my site, they will only see the intro once and have a header that
will play some background music without restarting while the visitor browses
the site? I want to stay away from using frames or iframes for search engine
and browser reasons

If you're dead set against frames, then you might want to look into Shared
Objects (basically Flash cookies). You could write an SO that indicates they've
seen the beginning of the movie once and jumpt to the appropriate place each
time the page reloads the movie.

Good luck!