First, in order to solve the problem, check if you plugged the speakers in the green jack of your computer. After that, go to Start, type dxdiag and open the utility. Click the Sound tab, check your audio card model and download and install the latest drivers from the developer's page. Also, right-click the Sound icon from System Tray and select Playback devices. Right-click anywhere in the provided window and select Show Disabled Devices. Click on your audio device, press Set Default and then hit Properties. Select Use this device (enable) as Device Usage, click Apply and then OK.
If the problem is not solved, launch your browser, press Ctrl + Shift + Delete key combination and clear all cache, cookies and temporary Internet files. After that, navigate to Adobe web page, select your operating system if it is not automatically detected and download and install the latest version of Adobe Flash Player. Also, make sure that the YouTube volume slider is not turned to Mute.