As of Firefox 4.0 and newer one needs both OSX 10.5+ and Intel hardware. You have PPC. http://www.mozilla.org/en-US/firefox/9.0/system-requirements/

Two options since you have PPC: either continue to use Firefox 3.6 (3.6.25 is current, released yesterday) which you have. http://www.mozilla.org/en-US/firefox/all-older.html

Or you can use a third-party build from http://www.floodgap.com/software/tenfourfox/ in order to have Firefox 9.0 though do note http://code.google.com/p/tenfourfox/wiki/PluginsNoLongerSupported

You can use Safari (which came with your OS) just fine. All things considered, because Apple no longer supports 10.4, and most plugin vendors (Java, Adobe) don't either, you're likely to have massive security holes that are unpatched for that reason.

In that sense, sticking to Firefox 3.6 shouldn't make you materially worse off than if you switched browsers or even if there were an update.

Does that mean that there is no way to avoid security holes until I buy a new computer with an intel processor? My MacMini OSX meets all my current needs reliably.

Thank you. I wish Apple had made it clear when I used Safari that it had security holes like Firefox notified me. Based on your info, I am now using TenFourFox as recommended by James. If I miss the plug ins I will buy a more modern computer.