You shouldn't have any difficulties - it does work OK on 10.6.8. The first thing to try is to download it again, from
http://get.adobe.com/flashplayer/
(only use this site, never install from any other site that offers it) and install it, making sure to quit all browsers before running the installer.
You can check that it has installed properly by going to the Adobe version test for Flash:
http://helpx.adobe.com/flash-player/kb/find-version-flash-player.html
If the installation fails, the next stage is to create a new admin user account on your Mac, and go through the same process. This would bypass anything in your user account which may be conflicting. Installing it in another user account will also make it available to your own account, so again if the version test works in the new account but not in the main one it is a conflict issue, though I don't know what might be causing it.