This free download software will help you: Moyea YouTube FLV Downloader which not only can download youtube videos but also other video-sharing web sites like fox, myspace, google video and so on.
It is easy to use:
1. Run the software
2. Click "Capture" tab
3. The program auto get URL
4. Click "Download" icon to start downloading.
5. Find the downloaded FLV on your PC by right clicking and choosing "Locate"
Get it here: http://www.flvsoft.com/download_flv
Here is a step by step video guide for you: http://www.youtube.com/watch?v=2rEL7_g7NrY
text guide with the screenshot: http://www.flvsoft.com/capture-youtube
Moreover, it is connected with FLV to Video Converter which helps convert flv to other video formats like avi, mpeg, mp4, mp3, wmv, mov, 3gp and more.
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More useful guides with the screenshots:
http://www.flvsoft.com/how-to-copy-youtube-files
http://www.flvsoft.com/how-to-grab-youtube
http://www.flvsoft.com/youtube-download-convert