Updated: October 11, 2013
The problem you're facing is as follows: Recently, you have upgraded your browser to version 24 or similar. Coincidentally, you have also updated your Flash Player. Ever since, all videos on Youtube are muted by default. You can hear audio after sliding the volume bar right in the Youtube player, but any subsequent attempt results in the same starting condition. All content is muted.
You have tried reinstalling your browser, your Flash Player, clearing browser data and cookies, you even signed in and out of Google to try to preserve settings, but nothing seems to help. Well, worry not, this tutorial will teach you all there is about getting your audio back. As always, it will be a unique tutorial unlike most of the guesswork out there.
So, the reason why your audio is muted is because there's been a change in how the Youtube and Flash combo works recently. If you do not let your Flash Player store content to your disk, then you will not be able to save any settings and all videos will start muted. This is an annoying new change, quite likely badly communicated. Indeed, if you check Chrome, you will find the exact same thing is in effect there. Muted, by default.
Now, what you need to do is open the Flash Player global settings menu. You can do it through the browser, as I've shown you in my lengthy tutorial on the subject, or you can fire up the desktop utility, akin to the one shown just below.
Under the Local Storage Settings tab, you will need to reset the saved options for ytimg.com. If you have blocked content, change to Ask me. If you already have the more permissive Ask me or Allow, you should be fine, but if not, then remove the storage and try accessing Youtube. Then, when you try to play a clip and slide the volume bar, you will get a popup like the one below, from the Flash Player, asking you to allow storing some local content to your computer.
Accept the request, and you will be fine. All your subsequent videos will play nicely with the correct audio settings. To use a meme, all your audio are belong to us. Now go away, enjoy life and thank Dedoimedo for being a friendly genius.
This is a very brief and totally mighty guide. First, I've refreshed your memory about the Flash Player settings. Second, we learned there's no reason to install and uninstall anything and start chasing waterfalls, I mean ghosts. Third, you know a little more about Flash local storage, cookies, content, how companies like to play silly tricks on you, and why multiple changes at the same time can be confusing.
Quite useful, would you not say. Enjoy your browser, your media and remember the basic rules. Start with simple things. Never doubt yourself, it's most likely some company or some site doing their usual share of nonsense. We're done. Let's get out of here.