I use a free program from the website called DVD videosoft. Just click on the first link below and you get reach the webstie.
There is a program called Free Youtube to iPod Converter and you can use it to convert Youtube videos into MP4 files. Same thing with the program called Free Youtube Download.
Lately, my computer hasn't been conpatible with downloading and installing Free Youtube Download, but with Free Youtube to iPod converter, so I would suggest using the Free Youtube to iPod converter.
And just because it says iPod converter doesn't mean it is compatible with other Mp3s. The format is just MP4 so don't worry about that.
The Free Youtube to iPod converter is very easy to use. It's just copy and pasting the link, and clicking download. Also, there are a lot of video quality settings. I would just download videos for Apple iPod and keep it at "MP4 - Original High Quality Youtube Download". Also, this program also comes with a Youtube to MP3 downloader if you look at the devices section. This converter is very fast in converting also.
Just follow the second link at the bottom to go directly to the downloading website.
There are all COMPLETELY freeware so they never ask to buy anything.
On the outside, the program doesn't look that good, but it functions really well, so I suggest you use that.
I never got a virus or any other malware from using this program, which makes it better. But I'm telling you now, if you download anything off the internet, you will someday get a virus. So just be safe.
Also, DVD videosoft has a whole bunch of other neat freeware programs and they don't take up much space on your computer.
(The size of each program is mostly about 3 MB when installed.)
For example, they have Youtube to MP3 converter which convertes videos into music files. There is free DVD burner, and there are about other 20 free programs. So enjoy.
Hope I was helpful :D I use this program myself all the time.