Chasing down why a PC's speakers do not work is a reasonably simple process, one that does not require any sophisticated tools. There are two diagnostic paths to pursue: verifying that the speakers themselves work, and ensuring that the PC's audio subsystem is putting out sound.
There are two types of PC speaker systems: ones that require an independent electrical power source, typically a AC/DC transformer, and speakers that need no external power.
Checking for Power
Examine the rear of your speaker set, paying close attention to the jacks. Do you have a power connector on the back of one of the speakers? If so, ensure that you have the correct power adapter for the speakers, as using the wrong AC adapter may damage the speakers.
Test the power supply by plugging the AC adapter and turning on the power switch. If you have one, is the power LED on the speakers lighting up? If not, you have identified the problem.
Checking Audio Connections
Check the cabling between the right and left speaker (or subwoofer, if included) is connected and verify that the cable that goes from your speakers to your computer is correctly plugged in.
With the power on and the volume turned up, press one of your fingertips against the metal part of the 1/8-inch stereo jack that would normally be plugged into the back of your computer. Do you hear a humming noise that stops when you remove your fingertip? If so, the speakers are probably working; if not, they are likely dead.
Checking the PC's Audio Software Settings
The most obvious audio setting to check is the PC's volume control. All of the major operating systems provide a tiny speaker icon usually located in your system tray, where the time is displayed. Open the speaker icon by clicking on it and verify that the volume control slider switch is turned up and that there is no check mark in the Mute box.
Please note: if you do not have a volume control/speaker icon, there is a good chance that the computer itself will require professional servicing.
Checking the PC's Audio Output
The easiest method to ensure that your PC's audio subsystem is functioning properly is to substitute the speaker system with a "known good" set of speakers or headphones. Most modern PCs have three to six 1/8-inch jacks located in the rear of the system for audio input and output.
Open up your favorite media player or point your browser to a site like YouTube so as to begin sending audio through the system. Now plug the 1/8-inch male audio jack into one of the 1/8-inch female jacks in the rear of the computer. If you don't hear anything, move on to the next 1/8-inch female jack. Continue this process until you have run out of jacks or have found the one that works.
Nothing Worked -- Now What?
If you have completed all of the steps above and still have no sound, the chances are pretty good that the problem lies either in your operating system or your PC's hardware. In either case, this will require the services of a computer repair technician.