There are several options available for downloading YouTube video, some paid and some free. While some require you install a program to do it, I will show you how to use a website to download the videos online for free.
For this to work you have to have Java installed, which most of the computers have already installed, but if don't have it just go to Java and download then run the installer. You will also need the latest version of Java to be able to download the YouTube videos. Here is the link: http://www.java.com/en/download/index.jsp
Once you have installed Java, go to http://keepvid.com/
to download your YouTube videos. In the URL box paste the full link of the video from YouTube that you want to download as per the grey example in there.
After you've done this, click Download. Java applet will start running and at this point you may be asked to allow it to run for this website. Click Run, and if you want to use it in the future as well, tick to 'Always trust content from this publisher'.
Website will generate 3 types of files for you to download: 3GP which is a mobile phones format, FLV which is the Flash format they are stored on YouTube, and MP4 format which can be played in Windows Media Player or on MAC OS, so I would suggest you download this version. Also, you can select the resolution for your video, the highest resolution generating a bigger size file. After you decided which one to choose, click on it and the download link for your video will be generated and ready to be saved locally on your computer.