You need to download Adobe Flash Player, since Apple is no longer shipping flash installed on its' computers.
http://get2.adobe.com/flashplayer/
did you try turning it off and on again? (The computer)
There's no need to reboot.
When you install FlashPlayer it tells you to shut down all browsers,
when it's done - your browser should have Flash already.
If it's not - you did something wrong before.
OR, you can turn on the HTML5 setting in youtube.
I was gonna suggest that, can't believe nobody mentioned it earlier?