I have hardly used this pc its one year old - the sound has gone off for some unknown reason. I have Logitec speakers plugged into the light green jack at the rear of the pc. There is a power light to the speakers. When i put the mounse pointer over the speaker icon at the bottom right of the screen it says "No Audio Output Device is installed"
Sounds like you lost your Sigmatel audio driver somehow (assuming you have the original integrated audio, and not an upgraded sound card in one of your PCI slots).
Easiest thing to try is System Restore if you have a known date when the audio was working. Beyond that...
Go to Control Panel, Sounds and Audio, click on the Audio tab. Check the default devices and see if you can change them to Sigmatel.
If you can't, go to your Device Manager (instructions HERE for XP), and under Sounds and Audio look for a Sigmatel device listed. If it's there, or if there's anything with a ? or ! next to it, right click and uninstall it, then reboot your PC. That should get it reloaded properly again.
If you're still having trouble, you can download and install a fresh copy of the audio driver from below - just select your operating system, then Audio, then the Sigmatel driver.