By default, the Flash player in the repositories is a 32-bit version of Flash. This version does not work properly on 64-bit installations of Ubuntu 10.04.
Someone made a bash script to automate the process of installing Flash Player 10 to Ubuntu. All you have to do is to type or copy-paste the following in the Terminal (Applications -> Accessories -> Terminal):
chmod +x install-flash.sh
It will ask for your password. Just provide your password. Restart Firefox and enjoy your Flash Player 10!