Recording Software Plus

I like collecting music videos and I make my own dvds out of them. However, most of the videos I would like to collect, are only found on youtube.com.
I would like to know if there is a software that will capture full screen videos as they are being played, and also get the audio as well. Free software is a plus, but I would be willing to pay a little for something that works.
I have already done my own searching but have found nothing but software that only records audio from a microphone. Meaning, it was meant to make tutorials. This is not what I am looking for. I want to grab the audio from the internet webpage while recording the webpage at the same time.
Please don't recommend any music video pirating sites, unless you know of one better than piratebay.org