If you want to record a "YouTube clip" from the net, Quicktime will only record the screen but no sound. To marry the two you need to install a "virtual audio device". Download: Soundflower. Its a small program with a sunfower image. Run the Soundflower Installer.pkg then follow the instructions.
Once you re-open quick time you will find under audio a new option for audio: record with soundflower. There will be no sound while you are recoding, but the recorded clip will have sound.
Do not forget to switch later back once you are finished to use the built in speakers on your Mac.
Another option is to screen the video on your iPad - connect the iPad to the Mac via cable. Open quicktime on the Mac and choose record iPad. I think this should also work.
Hope you come right. Remember copy right issues for intellectual property.