Is there a program that records all audio output of your computer? I mean, other than getting a microphone and sticking it next to the speakers while I play something off the internet.

There is, yes.

Many recording programs, such as Audacity, have this ability. However, it generally relies on your sound card. Most integrated cards don't have this capability, and won't allow you to do it.

I believe the setting in Audacity is called "What U Hear" under recording device.