Youtube videos will not play under OSX 10.6.8 I get a black rectangle with the "f" icon, but it won't play when I click on it.

Apple stopped updating OSX 10.6.8 so is its Flash too out of date to run?
I am going with your comment that says:


  Youtube says my version of Safari is no longer supported.


YouTube is (if they haven't finished already) migrating from Flash to HTML5.  If you can't view videos, this is the reason.

What you need to do, is get a browser that will support HTML5.  TenFourFox is the the browser that will solve this problem for you.  It is a port of a recent FireFox build to older versions of OS X

I personally have used this browser on an XServe G4 with 10.5 and I am able to view YouTube videos.