The files YouTube plays back on their site won't play on an iPod directly, but if you download the files to your hard drive and convert them to an iPod compatible format, they will definitely play on your iPod. Depending on whether you have Mac OS X or Windows XP, there's are two different software tools that make putting YouTube videos on your iPod almost automatic.
Putting YouTube Videos on an iPod for Windows XP
If you're a Windows user, the software you need to put YouTube videos on your iPod is iTube.
The software prompts you to paste in the link for the YouTube page with the video you want to convert, then it downloads the video, automatically converts it for your iPod and adds the video to your iTunes library. The next time you connect your iPod to your PC, you can sync the video to your iPod.
Find the video you want to download and copy the URL.
Paste the URL in iTube and click Go!
The software locates the video, downloads the video, and automatically converts the file to an iPod compatible video. The whole process takes a few minutes to complete depending on how long the video is. When the conversion process is complete, the video will be in the Movies section of your iTunes library.
Putting YouTube Videos on an iPod for Mac OS X
If you're downloading YouTube videos to Mac OS X, you need an app called PodTube. The software works almost automatically and gives you a couple of options. There is a slight trick in that the video you want to download from YouTube needs to be your top window in Safari when you start PodTube.
PodTube gives you the option to Encode the video for your iPod as well as letting you toggle whether the software will add the finished file to your iTunes library or not, so if you simply want to play the video locally on your Mac, you don't have to have the file in iTunes.