There are a variety of ways to download YouTube videos to your computer with online services or software.
Online Services
Many online services are sketchy and unreliable. The Wikimedia Community can only recommend its own internal products or tools from reputable free and open source partners.
Consider Commons:Commons:video2commons as the internal Wikimedia product
KeepVid is an online service that allows people to download YouTube videos in FLV, MP4, WebM, or MP3 format.
FLV - FLV files can be viewed with FLV players such as Applian FLV Player and software that can play a variety of video formats such as VLC media player. A list of more software is here.
MP4 - MP4 files can be played with a these software.
WebM - WebM files can be played with these software.
Find what YouTube video that you want, right click with your mouse on the URL and click copy. Open up KeepVid and paste the link in the Download box. Click Download and when Java tries to load, let it. The website uses Java to generate the download links. Choose what format and video quality that you want and then right click on the link. Click on Save As and choose where you want to save your file.