There are many ways to record audio of the Internet. This way involves using a wire called a aux to aux. That is that it has two identical Aux 3.5mm plug-ins on either end.
The procedure is simple. Take one end of the wire and plug it into the Microphone jack on your computer and plug the other one in the Headphone or Speaker jack. I'm using the one's top front of my desktop.
Then open up Youtube or whatever video source you want to record (audio only) and open up, in my case, Audacity. Start playing the video and press the record button on Audacity. When finished recording, just press the stop button on Audacity and then go to File - Export - Export as MP3 (or your desired format).