To start, Please follow the Uninstall instructions posted here: Uninstall Flash Player | Mac OS
Download the offline installer posted at the bottom of the Installation problems | Flash Player | Mac page, in the 'Still need help' section.
If it won't download directly, right click and "Save as"… and make note of the save location
DO NOT mount the installer DMG or run the installer yet!
Empty the trash
Quit your Browser (Press Cmd+Q - Don't just close the window)
Go to: Mac HD/Library/Internet Plugins
Verify that the Flash Player.plugin file is not present. If it is, trash it.
Go to: Mac HD/Library/Application Support/Adobe/Flash Player
Verify that the Flash Player Install Manager folder does not exist. If it does, trash the whole folder
Empty the trash.
Make sure the 'Reopen windows when logging back in' check-box is NOT selected (This will "kill" all running processes)
Before launching any apps, mount the DMG and run the installer.
BEFORE opening any Flash content in Safari, open Safari>Preferences, and check the Security tab to make sure "Allow all other plugins" is checked.
Message was edited by: Maria Vargas - updated link to the offline installer - added "Make sure the 'Reopen windows when logging back in' check-box is NOT selected" to the Reboot section.
Message was edited by: Maria Vargas the location of the offline installer has changed. updated FAQ with the new location.