Just broke a pin in a cable for my subwoofer so my speakers don't work any more. I do have headphones, but I can't get the computer to recognize them. They make a sound when they are plugged in, but no audio. In the sounds and devices section, it also doesn't show anything. I am not sure if the pc does recognize headphones in this area though. Any thoughts?

Your audio output cannot sense what you have connected to it. There should be sound coming through to your headphones.

What connections do you have available? Are your outputs set to a 5.1 system or some such? I would change that to just a stereo output when using headphones and plug into the main out for where the (regular) speakers would normally go.

Can't choose anything in sound and options. It says there is no sound card detected. I don't know why, but I haven't used this computer for quite some time and recently re-installed the os. It is an old Sony Vaio.

and recently re-installed the os
Perhaps the sound card's drivers were not installed. Check device manager for any problem devices with yellow exclamation marks. If you have a good internet connection then unless you have a very modern or odd-ball sound card it is likely you can install the drivers directly from the new hardware wizard by letting it go the MS Update site when you choose to "Update Driver" after Right Clicking on its icon there. Failing that you may need to point the wizard to the correct source for the drivers.