Improving on the advice given in Convert YouTube Video to MP3 Audio:
There's a much easier way, if you're using Windows. There's a bit of set-up, but then after that recording will be a breeze.
1. Enable Stereo Mix. This is disabled in Windows out of the box, but you can activate it. Here are instructions:
Right click on Vistas sound>control panel and select "show deactivated"
You should see the stereo mix device appear. You can activate it, by right click>activate.
Right click on the speaker icon in the system tray.
Select "Recording Devices"
In the resulting dialog, make sure you have the "Recording" tab selected and in the middle of the empty area, right click and select "Show Disabled Devices"
(If you need more, look up Windows Stereo Mix on Google and you'll find lots of advice).
2. Install and configure Audacity.
First, download and install Audacity from http://audacity.sourceforge.net/
To save MP3 files, you will also need to download and install LAME. Instructions are here: http://audacity.sourceforge.net/help/faq?s=install&item=lame-mp3
Start up audacity, record a little audio, then select File -> Export as MP3. The first time you do this Audacity will ask you were to find LAME.
Note that you configure your MP3 settings in Audacity's file formats. Select Edit -> Preferences -> File Formats. Select a high bit rate if you're recording music (I use 128), lower if recording audio (I use 32). High bit rates mean larger file sizes.
Finally, select Select Edit -> Preferences -> Audio I/O. Select 'Stereo Mix' from the options in the 'Recording' dropdown. Audacity will now record whatever comes out of your computer's speakers. Select 'stereo' if you want to record YouTube videos in Stereo.
Now you're all set. You only need to do this once (though if you want to record using a microsphone, you'll have to go back into Select Edit -> Preferences -> Audio I/O and select a microphone).
Start the video. Start recording in Audacity. You'll see the song being recorded in the Audacity window.
To save it as MP3 select File -> Export as MP3.
Note: if you are recording a lot of songs at once (if you were, say, recording concerts from http://www.wolfgangsvault.com/concerts/ then you want to 'Export Multiple'.
First, record the entire concert.
Second, in Audacity, add a lable at the beginning of each song. Use the song title for the lable. (Select the location of the lable by clicking on the timeline, then click Project -> Add Lable at Selection and type your title in the box.
Then, third, select File -> Export multiple to export all the songs at once.
(Note that while the process I describe here is legal in Canada, it might not be legal where you live to record audio from the internet. It is not recommended that you share these recordings.)