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There're many free software to do the trick, you could find a handful of the best ones from below:
Keepvid - this tool is actually a free online app which works for YouTube, Dailymotion and more. There's no need to register or pay, just visit the site, input the URL of a video and hit "Download". However, it does not let you convert videos to another format, therefore you may also check sites like keepvid if needed.
Freemake Video Downloader - though this software only works for Windows, it's totally free and only requires installation on your computer. It's able to handle multiple video clips synchronously.
yes, you could try audials light. it's free and it does record youtube videos. and, apart from that you get some extra features like internet radio recording and listening, podcast recording, watching music tv and mobile syncing. but if you want the real deal, I would recommend that you use audials moviebox; this one is able to record from pretty much any source, be that youtube or netflix. it's pretty awesome and you also get the option to convert your videos to different formats (mp4, avi, 3gp, wmv, mpeg etc.) so that you can use them on your other devices, xbox, ps3, ipad, iphone and so on. this is not a freeware however but yeah, I guess if you are looking into investing a little bitof money into it, this is what I would recommend that you use as a video recorder (and if you wanna find out exactly how it works, they offer some step by step instructions here).