From Fedora Project Wiki

Enabling Flash Plugin

For i386

Unlike earlier releases, no special steps are required to install the proprietary Adobe Flash Player 10 on Fedora 10.

For X86_64

Select the YUM version of the Adobe Flash Player from Adobe's download page. This installed Adobe's software repository so that updates can be applied by the normal Fedora update process.

Next some helper files are needed in addition to the Flash Player:

su -c 'yum install flash-plugin nspluginwrapper.i386 \
    alsa-plugins-pulseaudio.i386'

You will need to authorize Adobe's key during the install.

After the installation, exit all copies of Firefox and to enable the plugin.

Checking the plugin

Following the installation for either architecture, start Firefox and type in the addressbar:

about:plugins

You should see a section similar to the following: