I'll admit I'm not an expert when it comes to computers, but I'm not a complete newbie either. In the system tray I have two icons:
Realtek HD Audio Manager and the standard Windows Volumes icon. I have opened both and there doesn't seem to be anything wrong with either of them. Both show that there is audio being output (the mixers hve activity) which makes me think that it's a hardware problem. Neither of the audio jacks works however, which seems strange to me.
After a bit more searching on Google I figured out what the problem was. I somehow managed to change my settings from Speakers to Realtek Digital Output. Not sure how that happened, but everything seems to be working again.