Hey! I Can't Hear My Youtube!
I'm going to assume that your PC delivers sound just fine outside of Youtube. No sound at all is a whole other question.
Any number of things can block Youtube audio, but one of these three fixes will probably do the trick. The first two apply to situations where you never hear audio with Youtube videos. Use the third if the problem only occurs with videos embedded into web pages on sites other than youtube.com.
First Fix: A certain registry string value may be missing or pointing to the wrong file. Rather than walking you through fixing this problem manually, I've simply created a .reg file that will fix it for you. Download and run fix youtube.reg, and give it permission to alter your Registry. (This download works better in Internet Explorer than Firefox.)
If you prefer to do it yourself and know how, the value is called wavemapper, and it should be located in the HKEY LOCAL MACHINE/SOFTWARE/Microsoft/Windows NT/CurrentVersion/Drivers32 key. It should contain the string msacm2.drv.
Second Fix: If that didn't work, try the K-Lite codec pack, and make sure you download the Full version. Once you have it downloaded and installed, select Start, All Programs, K-Lite Codec Pack, Configuration, Codec Tweak Tools. Check the Fix broken sound (Midi, WaveOut) option. Do I have to tell you to click Next, then Finish?
Third Fix: If Youtube videos play their audio just fine at youtube.com, but sound like silent movies (without the music) when they're embedded in someone else's web page or an email, visit the Adobe Flash Player Settings Manager.
The resulting page has what looks like a screen shot of a dialog box. That's an illusion--it's a real dialog box. You'll see a row of small tabs at the top of the dialog box; you should be on the second one from the left--the one that looks like a folder in front of a globe.
Check the Allow third-party Flash content to store data on your computer option. Then click another one of these little tabs--it doesn't matter which one. After the window rewrites itself, click the tab you started on and make sure that the option is still checked.