You didn't specify which distribution you are using. So I will provide the instruction for installing the tarball download from the adobe website.
Download the Linux version of flash in the tar.gz archive form.
http://www.adobe.com/products/flashplayer/
- Extract the tarball.
Right-click on it and select extract here or use the terminal.
- Create a directory called plugins inside the hidden Mozilla directory in
~/.mozilla
It's in your home folder. You can see it by pressing Ctrl + H (GNOME) or Alt + . (KDE)
- Move "libflashplayer.so" into the newly created plugins directory
You can drag and drop or use the terminal.
Next time you start Firefox flash will be installed and working.
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To update flash in the future, download the new flash player and replace the old "libflashplayer.so" with the new one. It's that simple.