Ever since I've upgraded my Dell D620 from an Intel Core Duo T2300E to a Core 2 Duo T7200 I've grown to be "64 bits conscious". As a result today I'll explain how to install Adobe Flash Player in CentOS 6.2 x86_64.
To install Adobe Flash Player on a RHEL 6.2 clone, you'll need to change to root, install Adobe's YUM repository, import the associated GPG key and install Adobe Flash Player for x86_64.
For a 64 bits CentOS 6.2 installation, run the following commands:
# rpm -ivh http://linuxdownload.adobe.com/adobe-release/adobe-release-x86_64-1.0-1.noarch.rpm
# rpm --import /etc/pki/rpm-gpg/RPM-GPG-KEY-adobe-linux
# yum install flash-plugin
If you're running i386 you can install Adobe Flash Player following the instructions I've laied out here.
Launch Mozilla Firefox and in the address bar enter about:plugins. An Shockwave Flash entry should be available.
If you point to System -> Preferences you'll notice that a new entry named Adobe Flash Player is now available.
You can test if the plugin is working by visiting http://www.adobe.com/software/flash/about or by firing up a random Youtube clip.