There are a lot of music video on YouTube, and I'm told that kids today mostly discover new music there. Having that music as a MP3 can be very useful (although if you want to put it in your music library you will have to use your desktop to save the MP3 to iTunes). Here is how you do it using your iPad/iPhone/iPod touch.
The following is a adaption and combination two articles on this site. We will be downloading and saving a YouTube video to your Camera Roll (Using YouTube video in Pages). Afterward we'll open the music in the video in an audio editor app (Editing Audio in a Video). Finally the audio is saved as a MP3 audio file, and we can save the it to Dropbox or share it with email.
Save the YouTube Video to Your Camera Roll
There are several apps that download YouTube videos in the App Store. We need one that will save them to the Camera Roll. I know TubeBox does this, your mileage with other apps may vary. If you find an app that will let you send the video via "Open In". That will work too.
Use TubeBox to download the video then save it to the Camera Roll. If you need specific instructions take a look at the article on "Using YouTube video in Pages".
Next we need to send that video to a audio editor app, in this case TwistedWave. TwistedWave because it allows us to save the audio as a mp3. To do this we have to send media from the Camera Roll to an app via "Open In".
Send the Video from the Camera Roll to TwistedWave
Now that our YouTube video is in our device's Camera Roll, the video is accessible from many apps. Lets use one (iTransfer) to send the audio of the YouTube video to a sound editor app, TwistedWave. If you need specific directions look at this article (Editing Audio in a Video).
Save as MP3
TwistedWave will create mp3 audio files using the iPad itself (not an outside server). Take the Share button on the upper, left. Under Format, select "MP3". Tap the back button marked "Export audio", then choose what you would like to do with the audio after the export.
You can send the mp3 via email. Save it to Dropbox. Open it in another app. Since you can't add the mp3 to your music library from the device, save it back to your desktop (Dropbox, Email or Send to iTunes). From there you can add the mp3 to your iTunes library, and it will behave just like any other song.