To check if Adobe Flash Player is running properly simply try accessing any Youtube video. If you can see it OK then Adobe Flash Player is working. If you can't, follow these instructions:
Go to your list of installed programs and look for 'Adobe Flash Player Plug-in'. If it's already there (as it probably is) uninstall it. Then RESTART YOUR COMPUTER. (That's in capitals because it's IMPORTANT!!!).
Then go here
http://get.adobe.com/flashplayer/
Remove the tick under 'Optional offer'. Click 'Install Now' and follow the instructions. (NB: Note that you'll probably have to double-click on a downloaded file. Don't expect it to simply install itself without doing that!).
Then try accessing Youtube again. You should then be able to view the videos (and you shouldn't keep getting messages about a missing plug-in).