Fedora 10 Users
Go here if you are a Fedora 10 user: http://hacktux.com/fedora/10/flash
OpenSource Flash Plugins
Fedora 9 features the Open Source Flash player, Swfdec. You can install the Swfdec plugin for Firefox using Yum.
sudo yum install swfdec-mozilla
Fedora 9 also provides Gnash.
sudo yum install gnash-plugin
Install Adobe Flash Using Yum
Alternatively, you can use Adobe's Flash plugin. To install the Adobe Flash plugin on Fedora 9 visit http://www.adobe.com/products/flashplayer/ and click Download Now. Then select YUM for Linux and download the provided RPM (adobe-release-i386-1.0-1.noarch.rpm).
Install the downloaded RPM which installs /etc/yum.repos.d/adobe-linux-i386.repo.
sudo rpm -i adobe-release-i386-1.0-1.noarch.rpm
Check that you can access the Adobe Yum repository.
yum --disablerepo=* --enablerepo=adobe* list
Next, remove the Fedora Swfdec and Gnash plugin to avoid conflicts.
sudo yum remove swfdec-mozilla gnash-plugin
Now, install the Adobe Flash Plugin
sudo yum install flash-plugin
Finally, add the Adobe plugin is in your $HOME/.mozilla/plugins folder.
mkdir -p $HOME/.mozilla/plugins
cd $HOME/.mozilla/plugins
ln -s /usr/lib/flash-plugin/libflashplayer.so .
For Fedora 9 i386, you should be done. If you are running x86_64, read on.
Firefox x86_64 with 32-bit Adobe Flash
The Adobe Flash plugin is not available as an x86_64 package. If you need to run a 64-bit browser, but need Adobe Flash, you can use ndiswrapper to run the 32-bit Adobe Flash plugin on the 64-bit Firefox browser.
First, install Livna's release package for Fedora 9. This will provide you with access to Livna.org's Fedora 9 YUM repository using /etc/yum.repos.d/livna.repo.
rpm -ivh http://rpm.livna.org/livna-release-9.rpm
Next, install ndiswrapper from Livna.org.
sudo yum -y install ndiswrapper
Now, install both the i386 and x86_64 packages for nspluginwrapper from Fedora's YUM repositories
sudo yum -y install nspluginwrapper.{i386,x86_64}
Finally, install the i386 version of libflashsupport and the i386 version of pulseaudio-libs (for flash sound support).
sudo yum -y install pulseaudio-libs.i386 libflashsupport.i386
Sources
fedorasolved.org: Adding the Flash Player Plugin
adobe.com: Adobe - Flash Player: Installation instructions