U‘m using RHEL 6.x workstation with Mozilla Firefox 3.x. How do I install flash player so that I can use video site such as You-tube under RHEL 6.x or CentOS Linux 6.x 64bit operating systems?

You can install Adobe flash player by visiting the official Adobe site. You can download rpm file from the same site. For 64 bit system I recommend a preview release of flash player with native support for 64bit Linux.

Step #1: Download Flash Player

Visit any site such as youtube.com and you will see the following message on screen:

Click on Upgrade Now and you will see the screen as follows:

If you are a 32bit user select RPM format. All 64 bit version user should download a native version from Adobe labs:

Click on “Download Adobe Flash 11 beta 2”

Click on “Download plug-in for Linux 64-bit (TAR.GZ)” and save it to /tmp or your desktop. You can use the following wget command:
$ cd /tmp
$ wget http://download.macromedia.com/pub/labs/flashplatformruntimes/flashplayer11/flashplayer11_b2_install_lin_64_080811.tar.gz

Step #2: Install Flash Player

Type the following commands:
$ cd /tmp
$ tar -zxvf flashplayer11_b2_install_lin_64_080811.tar.gz
Sample outputs:

libflashplayer.so
usr/
usr/bin/
usr/bin/flash-player-properties
usr/share/
usr/share/pixmaps/
usr/share/pixmaps/flash-player-properties.png
usr/share/kde4/
usr/share/kde4/services/
usr/share/kde4/services/kcm_adobe_flash_player.desktop
usr/share/applications/
usr/share/applications/flash-player-properties.desktop
usr/share/icons/
usr/share/icons/hicolor/
usr/share/icons/hicolor/32x32/
usr/share/icons/hicolor/32x32/apps/
usr/share/icons/hicolor/32x32/apps/flash-player-properties.png
usr/share/icons/hicolor/22x22/
usr/share/icons/hicolor/22x22/apps/
usr/share/icons/hicolor/22x22/apps/flash-player-properties.png
usr/share/icons/hicolor/24x24/
usr/share/icons/hicolor/24x24/apps/
usr/share/icons/hicolor/24x24/apps/flash-player-properties.png
usr/share/icons/hicolor/16x16/
usr/share/icons/hicolor/16x16/apps/
usr/share/icons/hicolor/16x16/apps/flash-player-properties.png
usr/share/icons/hicolor/48x48/
usr/share/icons/hicolor/48x48/apps/
usr/share/icons/hicolor/48x48/apps/flash-player-properties.png
usr/lib/
usr/lib/kde4/
usr/lib/kde4/kcm_adobe_flash_player.so

First, close your browser. Type the following command to copy libflashplayer.so to /usr/lib64/mozilla/plugins/ as a root user:
# cp libflashplayer.so /usr/lib64/mozilla/plugins/
# chmod 0755 /usr/lib64/mozilla/plugins/libflashplayer.so
OR
# install -v -m 0755 libflashplayer.so /usr/lib64/mozilla/plugins
Open a webrowser and visit the following youtube video for testing purpose:
http://www.youtube.com/watch?v=u7deClndzQw

The instrctions were tested on the following:
$ lsb_release -a
Sample outputs:

LSB Version:	:core-4.0-amd64:core-4.0-noarch:graphics-4.0-amd64:graphics-4.0-noarch:printing-4.0-amd64:printing-4.0-noarch
Distributor ID:	RedHatEnterpriseWorkstation
Description:	Red Hat Enterprise Linux Workstation release 6.1 (Santiago)
Release:	6.1
Codename:	Santiago

Mozilla Firefox version:
$ firefox --version
Sample outputs:

Mozilla Firefox 3.6.20, Copyright (c) 1998 - 2011 mozilla.org

A Note About HTML 5

HTML5 is the fifth revision of the HTML standard (HTML4 was created in 1997) and as of August 2011 is still under development. However, many popular sites including youtube supports HTML5.
http://www.youtube.com/html5
From the page:

We support browsers that support both the video tag in HTML5 and either the h.264 video codec or the WebM format (with VP8 codec). These include:
=> Firefox 4
=> Google Chrome
=> Opera 10.6+
=> Apple Safari
=> Microsoft Internet Explorer 9
=> Microsoft Internet Explorer 6, 7, or 8 with Google Chrome Frame installed (Get Google Chrome Frame)