Reading the title may make you feel it's pretty simple - Download the flash player rpm package from adobe.com, install it and Done. I thought so too. But even if flash-plugin was installed, Firefox wasn't detecting it. So did some old-style googling, and came up with these commands.
Remember, my one is Fedora 14 x86_64, it might not work for other versions.
Step 1. Open Terminal and go to Root user mode
Step 2. Install Adobe YUM Repository RPM package
rpm -ivh http://linuxdownload.adobe.com/adobe-release/adobe-release-i386-1.0-1.noarch.rpm
rpm --import /etc/pki/rpm-gpg/RPM-GPG-KEY-adobe-linux
Step 3. Update Repositories(Optional, Do if you haven't done it recently)
Step 4. Install required packages
yum install flash-plugin nspluginwrapper.x86_64 nspluginwrapper.i686 alsa-plugins-pulseaudio.i686 libcurl.i686
This command should also work on Fedora 13, Fedora 12 and Red Hat (RHEL) 6 64-bit version (x86_64) (32 bit wrapped version)..
That's it. Restart Firefox and you should be able to run Flash stuff. Head for Youtube to test it.