YouTube plays videos directly from their website in a format called FLV. In order to get this video from the YouTube website and then burn it to a DVD, you'll need some software that can capture the video, convert it, and then burn it to your blank DVD. You can get software to do this for you online.
Download a youtube download program like YouTube Downloader, available free at download.cnet.com/YouTube-Downloader/3000-2071_4-10647340.html?tag=mncol. Click on the "Download Now" text on the main site, and then when a window pops up choose the "Save As" option to get the download going.
Open the installation file to install the program. Click "Next" on each screen until the program installs. Exit the installation screen and the program should start.
Visit a YouTube video you want to use. Right-click on the address at the top of your web browser and click "Copy." Go back into your YouTube Downloader program and you should see that the address has been automatically pasted where it says "Enter Video URL."
Click "OK." The program will download your YouTube video. Click on the circle next to "convert" and then click on the drop-down menu that says "Convert to" and select a video format like AVI." Click "OK."
Download and install a program like" Easy-MPEG-AVI-DIVX-WMV-RM-to-DVD." Insert your blank DVD-R and open the DVD program. Click on the green plus symbol on the right hand side of the program. A browse window will appear. Locate the video you just downloaded and converted and double-click on it.
Click the "Video to DVD" button to burn your video to DVD.